Pardon my terminology, but if x and y are both complex numbers, and there is a way "B" of bisecting the entire complex number space such that the B(xy) = B(x) * B(y), then what possible solutions are there for B?
For example, if B(x) = 0 when |x| = 0 or B(x) = 1 when |x| > 0, then if either x or y has a magnitude of 0, the product xy will have a magnitude of 0.
Are there any other solutions of B? If not, is it provable that B can only have this one example as a possible solution?