A complex number “is”† a quotient of two Euclidean vectors in a plane. That is, Z = u\v is the object which when you multiply it by one vector u, scales and rotates that into another vector v. We can write uZ = uu⁻¹v = v.

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

https://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

† More precisely, the algebra of complex numbers is structurally the same as the algebra of quotients of planar vectors.