Sketching the History of Hypercomplex Numbers
665 Brahmagupta (598-670) writes Khandakhadyaka which solves quadratic equations and allows for the possibility of negative solutions.
Abraham bar Hiyya Ha-Nasi (1070-1145) writes the work Hibbur ha-Meshihah ve-ha-Tishboret, translated in 1145 into Latin as Liber embadorum, which presents the first complete solution to the quadratic equation.
1484 Nicolas Chuquet (1445-1500) writes Triparty en la sciences des nombres. The fourth part of which contains the "Regle des premiers," or the rule of the unknown, what we would today call an algebra. He introduced an exponential notation, allowing positive, negative, and zero powers. In solving general equations he showed that some equations lead to imaginary solutions, but dismisses them ("Tel nombre est ineperible").
Nicolo Fontana (Tartaglia) (1500-77) finds the general method for solving all types of cubic equations and tells Cardano, under the promise that Cardano tell no one until he publishes first. Cardano tells everyone in 1545.
Girollamo Cardano
1545 Girolamo Cardano (1501-1576) writes Ars magna on the solutions of cubic and quartic equations. In it, solutions to polynomials which lead to square roots of negative quantities occur, but Cardano calls them "sophistic" and concludes that it is "as subtle as it is useless."
1572 Rafael Bombelli (1530-1590) publishes Algebra, making use of his "wild idea" that one could use these square roots of negative numbers to get to the real solutions by using a technique that later became known as conjugation. These techniques were originally written up in an early manuscript ca. 1550, but left unpublished.
1629 Albert Girard (1595-1632) publishes Invention nouvelle en l'algebre, stating clearly the relations between roots and coefficients, allowing of negative and imaginary roots to equations. (Girard's conceptualization of negative solutions also paves the way toward the idea of the number line, interpreting negative numbers as a kind of relative orientation.) Girard retained all imaginary roots because they show the general principles in the formation of an equation from its roots.
1637 Rene Descartes (1596-1650) coins the term "imaginary" for expressions involving square roots of negative numbers, and takes their occurrence as a sign that the problem is insoluble.
John Wallis
Gottfried Wilhelm Leibniz (1646-1716) revives some speculation into imaginary numbers, saying that they were a sort of amphibian, halfway between existence and nonexistence. [Nahin has the quote part as coming from 1702 about Bernoulli's trick of factoring polynomials into imaginary components.]
1673 John Wallis (1616-1703) publish his Algebra contains an early way to represent complex numbers geometrically.
1714 (March) Roger Cotes (1682-1716), in his paper "Logometria," deduces the formula, –iφ = ln[cos(φ) – i sin(φ)]. (Also republished in the posthumous Harmonia Mensurarum of 1722.) The result goes largely unnoticed.
1747 Leonhard Euler (1707-1783) shows that the logarithm of a negative number is imaginary.
Leonhard Euler
1748 Euler publishes Introductio in analysin infinitorum giving the infinite series formulations of exp(x), sin(x) and cos(x), and deducing the formula, exp(iφ) = cos(φ) + i sin(φ), though this has been discovered by Johann Bernoulli and others in different forms.
1749 Euler shows that a complex number to the power of complex number is also a complex number.

Jean le Rond d'Alembert's (1717-1783) construction of functions of a complex variable, obtaining what later is called the Cauchy-Riemann equations.
1777 Euler makes up the symbol i for Sqrt(-1)
1797 Caspar Wessel's (1745-1818) paper, "On the Analytic Representation of Direction: An Attempt," representing complex numbers graphically on a two-dimensional plane. (Printed in '98, and included in the memoirs of the Royal Academy of Denmark in '99, this work went largely unnoticed, not rediscovered until 1895. This plane representation today often is called the "Argand" or "Gaussian" plane.)
1806 Jean Robert Argand (1768-1822) publishes Essay on the Geometrical Interpretation of Imaginary Quantities, on the graphical representation of complex numbers.

Adrien-Quentin Buée (1748-1826) also publishes a paper on the geometrical representation of complex numbers.
1814 The memoirs of Augustin-Louis Cauchy (1789-1857) give the first clear theory of functions of a complex variable, though they are not published until 1825. Over the next three plus decades, Cauchy lays much of the foundation of modern complex function theory.
Augustin Louis Cauchy
1820 Siméon Denis Poisson (1781-1804) writes the first published example of integration in the complex plane (formulated by him in 1815).
1825 Cauchy publishes Memoire sur les integrales definies, prises entre des limites imaginaires. It contains his integral theorem with residues. (The period from 1825-50 is considered to be the beginning of modern complex analysis.)
1828 George Green (1793-1841) publishes a theorem relating contour and area integrals in the complex plane, often now called Green's Theorem. (Later rediscovered by Mikhail Ostrogradski (1801-62) in 1831.)

John Warren's (1796-1852) A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities.
1830 Augustus De Morgan (1806-1871) writes Trigonometry and Double Algebra, coming closer to the modern notion of mathematics as functional propositions, relating the rules of real numbers and complex numbers, though rejecting the possibility of triple or quadruple algebras.
1831 Cauchy shows that an analytic function of a complex variable can be expanded about a point in a power series in the neighborhood of the singularity. He formulates a rigorous algebra of complex numbers based on the geometry of the complex plane.

(April) Carl Friedrich Gauss (1777-1855) publishes his arithmetical theory of complex numbers (the term "complex" was coined by Gauss), allowing for a geometric interpretation of complex numbers. Gauss rigorously constructs an algebra of complex numbers, and points the way to what later becomes known as hypercomplex numbers.
1833 William Rowan Hamilton (1805-1865) first introduces a formal algebra of real number couples using rules which mirror the algebra of complex numbers.
1834 George Peacock's (1791-1858) Report on the recent progress and present state of certain branches of analysis, stating for the first time the idea that algebras should be constructed axiomatically and without reference to their intended interpretation.
1835 (June) Hamilton's Theory of Conjugate Functions or Algebraic Couples constructs a rigorous algebra of complex numbers as number pairs for the first time. Hamilton identifies x + iy with its coordinates (x,y) and rewrites the geometric definitions in algebraic form. (This had been discovered earlier by Gauss, but he never published the results.)
William Rowan Hamilton
1840 Duncan Farquharson Gregory's (1813-1844) "On the nature of symbolical algebra" points out that several different algebras can have operations which may be considered under one set of rules which would then apply to that class of algebras.
1841 De Morgan publishes "On the foundation of algebra," introducing symbolical algebra to explain the operations of specific algebras, considering +, -, *, :, unity, zero, commutativity, associativity, distributivity (not with those names), and gives a correct axiomatic treatment of equality.
1843 (October 16th) Hamilton thinks up the quaternion numbers. He defines a "vector" subspace, ai+bj+ck, elements of which can be interepreted as corresponding to a point in a 3-d Euclidean space. (See Links.)

(December 26th) John T. Graves writes to his friend Hamilton of his discovery of the octonions (also called "Cayley numbers"). Publication of the result, written up on Grave's request by Hamilton, waits until July of 1847.

Pierre Alphonse Laurant (1813-1854) and Karl Weierstrass (1815-1897) both derive extentions of Cauchy's theorem.
1844 Hermann Grassmann (1809-1877) begins work on his exterior algebra (through 1862), beginning with his work of this year, Science of Linear Extensions, dealing with the geometry of n-dimensional spaces. (See Links.)

Arthur Cayley (1821-1895) publishes, Chapters in the analytical geometry of (n) dimensions.

Hermann Grassmann
1845 Cayley describes the 8-dimensional octonions, called the Cayley numbers, which are both noncommutative and nonassociative, in an article "On Jacobi's elliptic functions, in reply to the Rev. B. Brouwin; and on quaternions." (Related info: see Links.)
1847 Ernst Kummer (1810-1893) introduces the concept of an ideal into number theory, a generalization of prime numbers that makes the fundamental theorem of arithmetic applicable to complex numbers.

George Boole (1815-1864) publishes The Mathematical Analysis of Logic, espousing a view of mathematics as a science of consistent symbolism, as opposed to numbers or magnitudes.

John Graves publishes "On algebraical triplets" for an algebra of the form a+be+ce2, where e3=-1, and shows that this algebra is the direct sum of the real and complex fields.
1851 Georg Bernhard Riemann (1826-1866) shows that any complex function can be one-to-one mapped onto a type of surface later known as a Riemann surface.
1852 James Joseph Sylvester (1814-1897) publishes "The proof of the theorem that every homogeneous quadratic polynomial is reduced by real orthogonal substitutions to a form of sum of positive and negative squares" (the law of inertia of quadratic forms).
Arthur Cayley
1853 Hamilton's Lectures on Quaternions, presenting his fully developed algebra and calculus, essentially showing that they also form a linear vector space over a real number field, introducing two notions of products over them, and showing the scalar (inner) product of two vectors was bilinear.
1854 Riemann's paper, "On the hypotheses that lie at the foundations of geometry," delivered at Göttingen (published by Dedekind in 1868), giving geometrical definitions of an n-dimensional manifold and introducing the notion of the curvature of a manifold.

Boole's Investigation of the Laws of Thought, establishing formal logic and a new (Boolean) algebraic system.
1858 Cayley's "A memoir on the theory of matrices" published in Philosophical Transactions, introducing matrices, their addition, multiplication, zero and identity, and a new interpretation of determinants. He shows that quaternions may be formulated in terms of matrices, and, generalizing a theorem on quaternions by Hamilton, formulated the Hamilton-Cayley theorem and proves it for matrices of order 2 and 3.
1859 Riemann extend's Euler's zeta function (Riemann's coining) to the complex domain.
1861 Karl Weierstrass proves that complex numbers are the only finite dimensional extension of the reals which preserves all the laws of arithmetic.
1862 Grassmann publishes a 2nd edition of his Science of Linear Extensions, being more accessible and popular among Biographies, elaborating on linear (in)depedence, subsets, intersections, and laying stress on the distinction between inner and outer products.
1866 Hamilton's Elements of Quaternions (postumous).
1867 Hermann Hankel (1839-1873) publishes Theory of complex numbers, containing a proof that complex numbers are the most general algebra that is possible under the fundamental laws of arithmetic, also helping to popularize Grassmann's ideas.
1870 Benjamin Peirce (1809-1880) publishes Linear Associative Algebras, one of the first systematic studies of hypecomplex numbers, working out the multiplication tables for 162 algebras. His son Charles S. Peirce shows that of all algebras of less than 7 dimensions, there are only three which are division systems, the real algebra, the complex and the quaternions.

(" 1870, the notion of the n-dimensional space Rn became the common property of the young generation that was forging ahead." — F. Klein, 1926)

Benjamin Peirce
1873 William Kingdon Clifford (1845-1879) developes a geometry of motion for the study of which he generalizes Hamilton's quaternions into the biquaternions (through 1876).
1878 Ferdinand Georg Frobenius shows, using algebraic topology, that quaternions are algebras which satisfy all properties of arithmetic save the commutative law of multiplication.
1879 Clifford publishes "Applications of Grassmann's extensive algebra" (Am. J. Math.) proposing a modification to Grassmann's algebra, formulating what is now known as a Clifford algebra.

Richard Dedekind (1831-1916) explicitly defines a number field.

William Kingdon Clifford
1881 Josiah Williard Gibbs (1839-1903) publishes his Vector Analysis (and work continues on this throughout the decade).
1884 Weierstrass publishes "On the theory of complex magnitudes formed of n principle units," showing that all commutative algebras without nilpotent elements is a direct sum of some number of copies of the real and complex fields.
Oliver Heaviside (1850-1925) writes Electromagnetic induction and its propagation, re-expressing Maxwell's electrodynamic theory, developing much of modern complex vector calculus and promoting its application in physics.
1886 Rudolf Otto Sigismund Lipschitz (1832-1903) rediscovers (independently) Clifford algebras, but is the first to apply them to rotations in Euclidean spaces.
1893 Fedor Eduardovich Molin (1861-1941) discovers that, up to an isomorphism, all complex simple algebras are full matrix algebras of order n. (Later independently rediscovered by Frobenius and Cartan.)
1896 Adolf Hurwitz (1859-1919) proves that every normed algebra with an identity is isomorphic to either the real, complex, quaternion or Cayley numbers.
1907 Joseph Henry Maclagen Wedderburn (1882-1948) proves that all simple associative algebras over a field P are precisely the full matrix algebras with elements from an associative division algebra over P.
1909 Edmund Landau's (1877-1938) book Handbuch der Lehre von der Verteilung der Primzahlen contains the earliest reference to the a + b i notation for complex numbers.
1957 John Willard Milnor, R. Bott and Kervaire show that if one relinquishes both commutative multiplication and the associative law, the complete set of arithmetic systems are the reals, the complex, quaternionic and Cayley numbers.
Sources: Allenby, R.B.J.T., Rings, Fields and Groups: An Introduction to Abstract Algebra, 2nd edition, 1991
Boyer, Carl B. , revised by Merzbach, Uta C., A History of Mathematics, 2nd edition, 1991
Kolmogorov, A.N., and Yushkevich, A.P. (eds.), Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory, 1992
Nahin, Paul J., An Imaginary Tale, 1998
Struik, Dirk J., A Concise History of Mathematics, 4th revised edition, 1987
The great MacTutor History of Mathematics Archive
Notes: In the 20th century, the focus switched over to algebras per se, associative or not, but this would take this timeline in a whole other direction. Perhaps at some point.