No. For a complex number $a$, let $f_a : \mathbb{C} \to \mathbb{C}$ be defined by $z \mapsto az$. Then the nice $n^{th}$ roots of $f_a$ are given by $f_b$ where $b^n = a$ and there's no canonical way to choose a root of this polynomial.
You can ask your question more generally: if $M$ is a monoid, when is it possible to define rational powers of elements of $M$ in a nice way? Depending on what you mean by "nice way" this is equivalent to $M$ being a vector space over $\mathbb{Q}$ (switching to additive notation). If you only want to demand that $n^{th}$ roots are unique and in addition $M$ is an abelian group, this is equivalent to $M$ being torsion-free, and if you only want to demand that $n^{th}$ roots exist (and in addition $M$ is an abelian group), this is equivalent to $M$ being divisible.