New answer to the generalized question. It's shown in previous answers that for $z^2$, and some other branched coverings, there are no immersions that are $C^1$-close immersionsexcept at the branch points. (I believe this should also imply that there are no immersions that are $C^1$-close except on a finite set.)
But $z^3:S^2\to S^2$ can be is arbitrarily $C^\infty$-approximated (on C^\infty$-close, except at the entire$S^2$, no need two branch points, to remove a finite set) by$C^\infty$immersions immersion in$\Bbb R^3$. (Also, any$C^\infty$map$S^2\to S^2$that is equivalent to$z^3$by a$C^0$change of coordinates is$C^\infty$-close on the entire$S^2$to an immersion in$\Bbb R^3$). To see this, pick a generic lift$f:S^1\to S^1\times\Bbb R$of the$3$-fold covering$S^1\to S^1$. It suffices to show that the composition$f':S^1\xrightarrow{f} S^1\times\Bbb R\subset S^2$bounds an immersion of a$2$-disk in a$3$-ball. Equivalently, we want to find a regular homotopy from$f'$to an embedding. But it is an exercise that that there are only two regular homotopy classes of immersions$S^1\to S^2$, distinguished by the parity of the number of double points (in the case of self-transverse immersions). 1 New answer to the generalized question. It's shown in previous answers that for$z^2$, and some other branched coverings, there are no$C^1$-close immersions. (I believe this should also imply that there are no immersions that are$C^1$-close except on a finite set.) But$z^3:S^2\to S^2$can be$C^\infty$-approximated (on the entire$S^2$, no need to remove a finite set) by$C^\infty$immersions in$\Bbb R^3$. To see this, pick a generic lift$f:S^1\to S^1\times\Bbb R$of the$3$-fold covering$S^1\to S^1$. It suffices to show that the composition$f':S^1\xrightarrow{f} S^1\times\Bbb R\subset S^2$bounds an immersion of a$2$-disk in a$3$-ball. Equivalently, we want to find a regular homotopy from$f'$to an embedding. But it is an exercise that that there are only two regular homotopy classes of immersions$S^1\to S^2\$, distinguished by the parity of the number of double points (in the case of self-transverse immersions).