This question is related to, but apparently not exactly the same as, Ramified cover of the $4$-sphere. Piergallini, *et al.* have singular points on their branch loci.

Which closed orientable $4$-dimensional manifolds can be realized as simple branched coverings of $S^4$ branched along a knotted surface? By *knotted surface*, I mean a smooth embedding of a closed orientable (but not necessarily connected) $2$-dimensional manifold. By *simple* I mean if the covering has $n$-sheets, then along the knotted surface the cover has $n-1$ sheets. Thus on the singular sheet the covering is $2$-to-$1$, as $z\mapsto z^2$.

I am especially interested in the case when $n=2$ or $3$ in which I can give an explicit immersion thereof into $S^4\times D^2 \subset of S^6$ in which the projection is the covering map.