**Initial question:**
I would like to know if there exists a sequence of $C^2$ immersions $f_k : S^2 \rightarrow \mathbb{R}^3$ which converge (in the $C^2$ topology) to $z^2$ except on a finite set of points, i.e. $f_k \rightarrow z^2$ in $C^2_{\text{loc}}(S^2\setminus \{ a_1, \dots , a_n \})$.

Above, $S^2$ is identified with $\hat{\mathbb{C}}$, the Riemann sphere. Hence, the function $z^2: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}} \sim S^2 \subset \mathbb{R}^3$ makes sense. In fact, my question is about any rational function $P/Q$ where $P$ and $Q$ are two elements of $\mathbb{C}[z]$; but we can start with $z^2$ in order to make it clearer.

The problem looks very hard topologically. For instance, if I assume "embedded" instead of "immersed", it is not very difficult to prove that such a sequence doesn't exist. But I am unable to show more.

**Further questions:**
Assuming there exists a sequence of immersions $f_k : S^2 \rightarrow \mathbb{R}^3$ satisfying the conditions in the first question, I would then like to know the following:

Can the immersions $f_k$ all extend to immersions from the closed ball to $\mathbb{R}^3$?

Can the sequence of immersions $f_k$ be chosen to have curvature bounded above?

How to produce such a sequence in the case of a general rational function? This problem reminds me of the sphere eversion: there appears to be no topological obstruction but it is hard to construct an explicit map.

I hope this is clear. Thanks in advance for your contribution.