That's pretty close to what Gauss said they should be called:

"If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question"

(https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-fo...)

I just think of them as different axes on a 2D Cartesian plane. That’s basically how they work.

EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.

But the quaternions sort of make the real component special dont they?

That it more an effect of how we use them in computer geometry. Unit quaternions with ijk plus real map to an xyz tilt plus a rotor. But I believe you could pick any such mapping.

On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation.

So in that case maybe there is something special about the real component.

Good and evil.

Spooky and strange?

Human and Werewolf.

Spooky Werewolf Algebras on the Monster group. Coming to arxiv this October.

I also like the name "lateral unit", maybe call the field "planar numbers"?