Originally Posted by

**Rockphed**
The simplest way to understand complex numbers is that i is a rotation from the real axis 90 degrees counterclockwise toward the imaginary axis. Extrapolating to 4 dimensions is going to make my head hurt, but let's try. I am going to assume that i,j,k, each behave as rotations in their own plane. But rotating i by j doesn't start on the j-plane, so rotating about an axis orthogonal to the j-plane can't get you onto the j-plane. Likewise, it can't get a real number or the i-plane and j-plane would be congruent, so it has to end up on the k-plane. It works out to be a bit cross-product like (albeit possibly with commutation), so ij=k, jk=i, and ki=j.