  A non-Hermitian generalisation of the Marsden--Weinstein reduction method is
introduced to construct families of quantum $\mathcal{PT}$-symmetric
superintegrable models over an $n$-dimensional sphere $S^n$. The mechanism is
illustrated with one- and two-dimensional examples, related to $u(2)$ and
$u(3)$ Lie algebras respectively, providing new quantum models with real
spectra and spontaneous $\mathcal{PT}$-symmetric breaking. In certain limits,
the models reduce to known non-Hermitian systems and complex extensions of
previously studied real superintegrable systems.
