On the complex modulus page from Wolfram MathWorld, I find an interesting statement:
The only function satisfying identities of the form $$\lvert f(x + iy) \rvert = \lvert f(x) + f(iy) \rvert$$ are $f(z) = Az$, $f(z) = A\sin(bz)$, and $f(z) = A \sinh (bz)$.
It refers to a paper that I cannot find on the Internet. It is trivial to show that these functions satisfy the identity, but why are they the only ones?
The mathworld entry seems to contain some errors.
First, the title of the paper is wrong, it should be "A curious trigonometric identity". The paper available for free on jstor here. The general idea of the proof is to replace the function by its MacLaurin series.
Second, it is missing the critical hypothesis that $f(z)$ is regular for $|z|<r$ (presumably for any positive value $r$).