It is obvious that $\left|e^{v}\right|=e^{\text{Re }v}>0$ showing that $\ln z$ is not defined for $z=0$ .
So the expression $z^{u}=e^{u\ln z}$ cannot be used here.
Nevertheless we don't hesitate to say things like: $0^{3}=0\times0\times0=0$.
Are there some conventions here?
Actually, it is not obvious that $|e^v|=e^{\operatorname{Re}v}$. It is true nonetheless. And the fact that $\ln z$ is not defined when $z=0$ is more basic than that. In fact $\ln z$ is a number such that $e^{\ln z}=z$. But we never have $e^v=0$, since $e^v.e^{-v}=1$.
The concept of $a^n$, when $n\in\mathbb{N}$, is more elementary than the concept of exponential function. It just means$$\overbrace{a\times a\times\cdots\times a}^{n\text{ times}}.$$It turns out that, when $a\neq0$, this is equal to $e^{n\log a}$, where $\log a$ is any logarithm of $a$. But we don't need this to define $a^n$. And, by the definition that I mentioned, it is clear that$$(\forall n\in\mathbb{N}):0^n=0.$$