A hypercomplex number system is an finite-dimensional algebra over the field $\mathbb{R}$ of real numbers. A hypercomplex number is an element of one of these algebras.
There is only one? hypercomplex number system of dimension $0$, of course.
There is also only one hypercomplex number system of dimension $1$, which is $\mathbb{R}$ itself.
Up to isomorphism, there are three hypercomplex number systems of dimension $2$, each of which is a commutative algebra. Since they each have one nontrivial automorphism (an involution called conjugation), there are only $3/2$ of these algebras by groupoid cardinality. Given the rule
(for $a, b$ fixed real numbers), the algebra $\mathbb{R}[e]$ may be identified as one of these three cases by the sign of $b^2 - 4 a$. They are:
Famous hypercomplex number systems of dimension $4$ include the quaternions and the bicomplex number?s.
Of course, these are not the only possibilities by any means. One can always form the direct product of two hypercomplex number systems to get a hypercomplex number systems with the sum of the dimensions. Another way to double the dimension is to form the tensor product with any of the hypercomplex number systems of dimension $2$; in particular, complexification (the tensor product with the complex numbers) is often denoted by the prefix ‘bi‑’. The Cayley–Dickson construction will double the dimension of any hypercomplex number system equipped with a (possibly trivial) involution. Another way to generate associative hypercomplex number systems is through Clifford algebras.
There is a thorough list of examples on the English Wikipedia.