Quote Originally Posted by Rockphed View Post
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.
Which isn't a surprise as basically he wanted vectors and cross products 40 years too early.

I looked it up, and I was right historically i*j*k=-1 is axiomatic and written on Brougham bridge. Although 3B1B's presentation of complex numbers is based on a more modern thinking of what they mean, so he'd probably start by constructing the rotational analogue and then derive the historical axioms as the interesting parallel rather than the other way round. A third option would be to start with the multiplication timeline.

Again as Rockphed implies, many of the other options have the wrong sort of consequences.
If i,j,k aren't symettric then that's a bit messy (though you do have Tessarines, in this case you have values such that a*b=0).
If the orthoganal 'vectors' were unaffected by the rotation then the resultant 'direction' is basically dominated by the last two numbers (I think), etc... (which I'm sure you could do something with but isn't a good 4D complex number)