Let $f:C\to C$ and if such
$$|f(x-y)|=|f(x)-f(y)|,\forall x,y\in C$$
Then is it true that $$f(x+y)=f(x)+f(y),\forall x,y\in C$$

I have prove when $f:R\to R$.But for complex,I can't solve it.It is not difficult to prove the real number, but I think it is not easy to solve the plural situation.and this problem is from a paper conjecture