2 added 3 characters in body

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

A friend of mine is looking for a connected 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 $C$?

1

Riemann surface disconnected at infinity

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

A friend of mine is looking for a connected (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 $C$?