Fix $n$ and let $H_1$ and $H_2$ be two hypersurfaces in ${\bf A}^n$ (not necessarily smooth or irreducible, but we'll assume reduced). If the complements $U_i = {\bf A}^n \setminus H_i$ are isomorphic as schemes, does this imply that the $H_i$ are isomorphic?

From asking other people, the answer is yes via Euler characteristic arguments when the $H_i$ are smooth, but I have in mind singular examples.

As for the ground field, the polynomials I have in mind are defined over ${\bf Z}$, so choose whatever field you like. Probably I'd prefer an answer for the complex numbers though.