It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the function), is locally conjugate (by right composition with a local biholomorphism) to a polynomial (in fact, a finite jet of the function).
This statement is not true anymore when $n\geq 3$ if the singularity is not isolated, although I can't find a reference containning examples of germs not locally conjugate to polynomials. Yet if I remember correctly, it is true when $n=2$.
Also I would be interested in knowing what happens in the meromorphic case, which I believe is still open.
I would be grateful for any pointer in the litterature towards such results!
PS: I hope this question is not a duplicate. I couldn't possibly browse through all the search results for requests like «polynomial holomorphic function», and was unable to think about sharper keywords…