The hypercomplex numbers are a generalization of the complex numbers. They were created in an attempt to describe certain geometric operations in spaces with a dimension higher than the 2-dimensional plane. Operations on the complex numbers can be used to describe many of the geometric operations on the plane. For instance multiplication by a real number corresponds to a scaling of the plane. Adding complex numbers corresponds to translation of the plane.
More examples are available on the geometry and imaginary numbers page.
It is known now that this is not possible. The only dimensions in which there are hypercomplex numbers which allow for a notion of division are dimensions 4 and 8. In both cases these number systems are unique - the only numbers with these geometric properties in dimension 4 are the Quaternians and in dimension 8 are the Cayley Numbers. Also, with increases in dimension, we can not maintain some of the properties which make number systems simple. In both the Quaternians and the Cayley numbers, multiplication is non-commutative.