# Multiplicative but non-additive function $f : \mathbb{C} \to \mathbb{R}$

I'm trying to find a function $f : \mathbb{C} \to \mathbb{R}$ such that

1. $f(az)=af(z)$ for any $a\in\mathbb{R}$, $z\in\mathbb{C}$, but
2. $f(z_1+z_2) \ne f(z_1)+f(z_2)$ for some $z_1,z_2\in\mathbb{C}$.

Any hints or heuristics for finding such a function?

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HINT: Look at $z$ in polar form.