  We study the complement problem in projective spaces $\mathbb{P}^n$ over any
algebraically closed field: If $H, H' \subseteq \mathbb{P}^n$ are irreducible
hypersurfaces of degree $d$ such that the complements $\mathbb{P}^n \setminus
H$, $\mathbb{P}^n \setminus H'$ are isomorphic, are the hypersurfaces $H$, $H'$
isomorphic?
  For $n = 2$, the answer is positive if $d\leq 7$ and there are
counterexamples when $d = 8$. In contrast we provide counterexamples for all
$n, d \geq 3$ with $(n, d) \neq (3, 3)$. Moreover, we show that the complement
problem has an affirmative answer for $d = 2$ and give partial results in case
$(n, d) = (3, 3)$. In the course of the exposition, we prove that rational
normal projective surfaces admitting a desingularisation by trees of smooth
rational curves are piecewise isomorphic if and only if they coincide in the
Grothendieck ring, answering affirmatively a question posed by Larsen and Lunts
for such surfaces.
