It is not affine transforms per se but rather the expansion into homogeneous coordinates that enables translation by treating it as if it's a shear that leaves the reciprocal dimension untouched.

> Rotation = multiplying by an imaginary unit.

This is also not quite right.

Rotation is multiplying by a complex number with a magnitude of 1 (or perhaps you meant to say "raising a number to the power of i"?)

Sorry I meant "complex unit".

"complex number with a magnitude of 1" is the definition of a "complex unit".

I like "roots of unity" (not necessarily rational) or "unit phasor" or "non-integer powers of -1"