  We investigate Hardy-Rellich inequalities for perturbed Laplacians. In
particular, we show that a non-trivial angular perturbation of the free
operator typically improves the inequality, and may also provide an estimate
which does not hold in the free case. The main examples are related to the
introduction of a magnetic field: this is a manifestation of the diamagnetic
phenomenon, which has been observed by Laptev and Weidl in \cite{LW1999} for
the Hardy inequality, later by Evans and Lewis in \cite{EL2005} for the Rellich
inequality; however, to the best of our knowledge, the so called Hardy-Rellich
inequality has not yet been investigated in this regards. After showing the
optimal inequality, we prove that the best constant is not attained by any
function in the domain of the estimate.
