I want to find counterexamples for the following "states":
if $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad \forall x ,y\in\mathbb{C}.$$
prove or disprove $$f(x+y)=f(x)+f(y),\forall x,y\in\mathbb{C}?$$ Can you give me a hint of what examples may work?
Thank you.