Compatible notion of degree for $z^k : S^1 \to S^1$, when $k$ is not an integer?

Question

I'm thinking about degree as the induced map on the first homology groups - the degree of $z^k$ is $k$, when $k$ is an integer. What happens when $k$ is not an integer? Is there a compatible notion of degree, taking values in $\mathbb{C}$?

(My complex analysis is not that developed yet, so maybe this is a stupid question.)

Answer 1

When $k$ is not an integer, then there is not a continuous map "$z^k$" from $S^1\to S^1$.