I've always been taught that one way to look at complex numbers is as a cartesian space, where the "real" part is the x component and the "imaginary" part is the y component.
In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.
However, is there a geometric interpretation for the multiplication of two complex numbers?
I tried out two test ones, $3+i$ and $-2+3i$, which multiply to $-9+7i$. But no geometrical significance seems to be found.
Is there a geometric significance for the multiplication of complex numbers?