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?

B is defined as a boolean function on a complex number, $B : \mathbb C \to \mathbb B$, so that B bisects the complex numbers into two regions such that $B(z_1 z_2) = B(z_1) B(z_2) $. What are the possible solutions for $B$?

One such solution is $B(z) = \begin{cases} 0, {if} |z| = 0 \\ 1, {otherwise}\end{cases}$, so that if either $z_1$ or $z_2$ has a magnitude of 0, the product $z_1 z_2$ will have a magnitude of 0, and the condition for $B$ will be met.

Are there any other solutions of $B$? If not, is it provable that $B$ can only have this one solution above?