  We provide explicit counterexamples to the so-called Complement Problem in
every dimension $n\geq3$, i.e. pairs of non-isomorphic irreducible
hypersurfaces $H_1, H_2\subset\mathbb{C}^{n}$ whose complements
$\mathbb{C}^{n}\setminus H_1$ and $\mathbb{C}^{n}\setminus H_2$ are isomorphic.
Since we can arrange that one of the hypersurfaces is singular whereas the
other is smooth, we also have counterexamples in the analytic setting.
