You want geometric algebra. In 2D GA, you would have two unitary vectors, i and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of between them (or their division) would be a bivector: i * j = ij = - j * i. You rotate 90 degrees the i and j vectors using the bivector.

The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.

That does sound pretty close to what I had in mind. Are i and j from your example associated with orthogonal axes in a plane?

Yes, they are. i would be the unitary vector in the horizontal direction, x, and j in the vertical one, y.

In fact, geometric algebras are very general, and the one I briefly sketched is not the only possible interpretation. If you are interested, you can find many good introductions online, directed at different audiences. You can also search for the term "Clifford algebras" if you are interested in a more formal approach.

Reduce these to the integral powers of i: i^0, i^1, i^2, i^3. The symmetries occur two apart (0, 2; 1, 3). Higher integral powers are equivalent to the 4 listed, i^n = i^(n mod 4).

Really, each power corresponds to an additional 90 degree rotation. 0th = 0 degree rotation (just scaling), 1st = 90 degree rotation, 2nd = 180 degree rotation, 3rd = 270 degree rotation (equivalently -90 degree). So that seeming CW rotation is also 3 CCW rotations.

Sure, I guess that's a consequence of general rotational symmetry.

The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.

Separate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.

But none of that is related to my key point: multiplication by unitary values is significantly different depending on which axis the unit comes from (real vs imaginary), hence my usage of 'asymmetric'.

In the last part of my first comment I point out the benefit (that I perceive) arising from the asymmetry. I also still wonder if a symmetric version (like I describe in the first comment) would be possible.

Splitting the geometric behavior into rotation, reflection, and scaling is interesting—so thanks for the description—I just don't see how it relates to my comment.

It would be great to get this level of explanation extrapolated all the way out to describe the behavior of octonions. Kidding, but not really ;)