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
(for fixed real numbers), the algebra may be identified as one of these three cases by the sign of . They are: * the complex numbers (elliptic case, ), * the dual numbers (parabolic case, ), * the perplex numbers (hyperbolic case, ).
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.