[ogogmad]

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

Similar story for other 2D number systems:

For the dual numbers, they express scalings and "Galilean" rotations (i.e. shears).

For the double numbers, they express scalings and "Minkowski" rotations (i.e. Lorentz boosts).

Unfortunately, some of the nice theory of the complex numbers doesn't generalise easily to the dual numbers or double numbers. I'm thinking specifically of complex analysis which is very, very nice, and much nicer than real analysis. But I think these planar number systems have their own intriguing character: For instance, see "screw theory" and "automatic differentiation" for two distinctive applications of the dual numbers.