Solve $|z|=\arg z$

Interpret ${\rm arg}$ as principal value ${\rm Arg}$, and write $z$ in the form $$z=r(\cos\theta+i \sin\theta)\qquad(r\geq0,\quad -\pi<\theta<\pi)\ .$$ Your condition then amounts to $$r=\theta\ ,$$ so that necessarily $r=\theta>0$, since $\theta$ is undefined at $z=0$. This shows that the set $$S:=\bigl\{z=\theta\,(\cos\theta+i\sin\theta)\in{\mathbb C}\>\bigm|0< \theta<\pi\bigr\}\ ,$$ an arc of an Archimedean spiral

is the solution to your problem.