This is Exercise EP $8$ from Fernandez and Bernardes's book Introdução às Funções de uma Variável Complexa (in Portuguese).
If $z,w\in \mathbb{C}^{\times}$, then $|z+w|=|z|+|w|$ iff $w=tz$ for some $t\gt 0.$
I supposed that $|z+w|=|z|+|w|$ and by squaring it I got $\operatorname{Re}(z\bar{w})=|z||w|$ and I am stuck. I would appreciate a hint in this question.