  We consider non-Hermitian but PT-symmetric extensions of Calogero models,
which have been proposed by Basu-Mallick and Kundu for two types of Lie
algebras. We address the question of whether these extensions are meaningful
for all remaining Lie algebras (Coxeter groups) and if in addition one may
extend the models beyond the rational case to trigonometric, hyperbolic and
elliptic models. We find that all these new models remain integrable, albeit
for the non-rational potentials one requires additional terms in the extension
in order to compensate for the breaking of integrability.
