Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higherdimensional Euclidean space (4 dimensions for the quaternions, 8 for the octonions, 16 for the sedenions). More precisely, they form finitedimensional 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 CayleyDickson construction. The Clifford algebras are another family of hypercomplex numbers.
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