A family of hypercomplex numbers is introduced in which multiplication is commutative and members can have up to eight components. In particular, the eight basis elements {E} contain those for ordinary complex numbers, E**=E, as well as new elements where E**=−E; the operation * being the generalization of complex conjugation. This family lends itself to the description of quantum mechanical spin states in that it offers a simple treatment of time reversal, representations with the same conjugation properties as underlying operators, and explicit continuous-angle spherical harmonic functions Zsm(θ,φ) analogous to the Ylm(θ,φ) for orbital angular momentum. The new elements are especially well suited for half-integral spin states, whereas conventional complex numbers remain useful for integral spin states.