Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, 8 for the octonions, 16 for the sedenions). More precisely, they form finite-dimensional algebras over the real numbers. But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed - see fundamental theorem of algebra.

The quaternions, octonions and sedenions are generated by the Cayley-Dickson construction. The Clifford algebras are another family of hypercomplex numbers.