It is well known that for $x\in \mathbb{R}$ we have $$ e^{x} = \lim_{n \to \infty} \left(1+ \frac{x}{n}\right)^n. $$ This follows quickly by considering logarithms and using L'Hospital's rule. However, for $z \in \mathbb{C}$ this would involve taking complex logarithms. I am not convinced that this proof still works and I was wondering if there is another simple proof of this fact when $z \in \mathbb{C}$.
EDIT: For definition I am using the Taylor series $$e^z := \sum_{k=0}^\infty \frac{z^k}{k!}.$$ I am okay with all the standard alternative definitions and properties of $e^x$ when $x \in \mathbb{R}$, so feel free to use those if the real case can somehow be extended to the complex case.
See the Wikipedia entry on characterizations of the function $e^x$ for $x \in \mathbb R$, and suggestions for extending each of these characterizations to a larger domain.