There is only one hypercomplex number system of dimension , of course.
There is also only one hypercomplex number system of dimension , which is itself.
Up to isomorphism, there are three hypercomplex number systems of dimension , each of which is a commutative algebra. Since they each have one nontrivial automorphism (an involution called conjugation), there are only of these algebras by groupoid cardinality. Given the rule
e^2 = a + b e
(for fixed real numbers), the algebra may be identified as one of these three cases by the sign of . They are:
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 ; 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.