complex geometry - Riemann surface disconnected at infinity

This question may be trivial, I did not think hard about it.

A friend of mine is looking for an irreducible (reduced) analytic subspace $C subset mathbb{C}^2$ with the following property. Let $f colon C rightarrow mathbb{C}$ be the projection on the first factor. He wants that

1) All singular points of $C$ and all ramification points for $f$ lie in a limited set, so removing that set we obtain a topological covering from some open set of $C$ to $mathbb{C}$ with a ball removed.

2) That covering should be trivial (even better if it is finitely-sheeted).

So the curve $C$ is connected, but only if one passes near the origin. Sufficiently far from that ther should be no way to jump between sheets. Is it possible to find such a $C$?

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