Let $\text{Arg}(z):\mathbb{C}\to(-\pi,\pi]$ be the argument function. Find all the values of $z\in\mathbb{C}$ such that $\left|z\right|e^{i\text{Arg}(z)}=z$.
My initial intuition was that the above is true for every $z\in\mathbb{C}$; However something tells me that might not be true, since that would make the original question I had to use this identity for much easier. Nevertheless, I could not figure out which values would be problematic.
Thank you very much!