False proof that $ρe^{iθ} = ρ$ and so complex numbers do not exist?

The error lies in assuming that $(\forall a,b\in\mathbb{C}):e^{ab}=(e^a)^b$.

It's worse than wrong; it doesn't make sense. The reason why it doesn't make sense is because $e^a$ can be an arbitrary complex number (except that it can't be $0$). And what is $z^w$, where $z,w\in\mathbb C$? A reasonable definition is that it means $e^{w\log z}$, where $\log z$ is a logarithm of $z$. Problem: every non-zero complex number has infinitely many logarithms: if a number $\omega$ is a logarithm, then every number of the form $\omega+2k\pi i$ ($k\in\mathbb Z$) is also a logarithm.

Because for complex numbers, $e^{zc}\ne(e^z)^c$ for $z,c \in \mathbb{C}$.