Integer powers in the complex field

Fix $z\in\mathbb C$. Apply The recursion theorem to the set $\mathbb C$, the initial value $1$, and the function $w\mapsto wz$. This defines the natural powers uniquely by

$$z^0=1,\qquad z^{n+1}=z^nz\quad(n\in\mathbb N).$$

Let $\jmath\colon\mathbb N\to\mathbb Z$ be the embedding of The naturals embed in the integers. For an integer $r\ge0$, that lemma gives a unique $n\in\mathbb N$ with $r=\jmath(n)$; define $z^r:=z^n$. If $r<0$ and $z\ne0$, there is a unique $n\in\mathbb N$ with $n\ge1$ and $-r=\jmath(n)$; define

$$z^r:=(z^n)^{-1}.$$

The inverse exists because $\mathbb C$ is a field by The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$. Thus nonnegative integer powers are defined for every $z$, while negative integer powers are defined exactly when $z\ne0$; in particular $0^0=1$ and no negative power of $0$ is defined. The integer and its unique natural representative are never conflated.