How are powers of complex numbers defined? Suppose I have some number $z\in\mathbb{C}$. It makes sense that there are $n$ unique solutions to $w^n=z$. Where we define this power in terms of complex multiplication.

How is this extended to $z^w$ where $w\in \mathbb{C}$. I assume some sort of limiting process is required from natural to rational to real to complex? Is there a good resource for finding where this is defined?