  In \cite{DJL07} it was shown that if $\scra$ is an affine hyperplane
arrangement in $\C^n$, then at most one of the $L^2$--Betti numbers
$b_i^{(2)}(\C^n\sm \scra,\id)$ is non--zero. In this note we prove an analogous
statement for complements of complex affine hyperurfaces in general position at
infinity. Furthermore, we recast and extend to this higher-dimensional setting
results of \cite{FLM,LM06} about $L^2$--Betti numbers of plane curve
complements.
