Let $f:X \to Y$ be a surjective finite morphism of schemes over $\mathbb{C}$. We know $f$ must be quasi-finite, i.e., its fibers are finite. We also know the cardinality of the finite fiber above a generic point $y$ of $Y$ is the same, which is the degree of $f$.
My questions are:
Is $f$ always a branched covering? If not, what's a counter-example?
What if we further require that $f$ is an étale morphism?
Edit:
I'm not sure if it always makes sense to talk about branched coverings in the full category of schemes over $\mathbb{C}$. We can take the schemes to be algebraic varieties for now. By a branched covering $f:X \to Y$, I mean a morphism $f:X \to Y$ such that it's indeed a covering map over an open $V \subset Y$.