Let $\mathbb{C}(t)$ be the field of rational functions $f(t) = \frac{p(t)}{q(t)}$ with $p,q\in\mathbb{C}[t]$.

For instance, the function $g(t) = \sqrt{t}$ does not belong to $\mathbb{C}(t)$ but is lies in its algebraic closure $\overline{\mathbb{C}(t)}$ since it is a zero of the polynomial $X^2-t\in\mathbb{C}(t)[X]$.

Question.Is it correct to think of elements of $\overline{\mathbb{C}(t)}$ as functions $h = h(t)$ with values in $\mathbb{C}$ such that $P(h(t)) = 0$ for some $P\in \mathbb{C}(t)[X]$?