For example, complex numbers in the eigendecomposition of a real-valued square matrix encode permutations among dimensions in a particular eigenspace via the roots of unity — which amount to rotations. So they’re a bookkeeping device that allows the eigendecomposition of an automorphism of a real vector space to actually work in all cases.
Complex number is a natural fit for rotation because operations of unit complex numbers end up on the same complex unit circle/sphere. e.g. Multiplying two quaternions performs two rotations one after another.
Of course, there's the octonion complex number with 8 axes for representing rotation in 4D hyperspace.
[1] https://en.wikipedia.org/wiki/Discrete_Fourier_transform
[1]http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.h...
- points on a plane
- points on a sphere (via one-point compactification)
- points on the hyperbolic plane
The last is linked to the Poincare half-plane model and the Poincare disk models of hyperbolic geometry.