The rational Calogero model based on an arbitrary rank-n Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on S n−1 subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a PT-symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The PT deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic rank-two (A 2, G 2) and all rank-three Coxeter systems (AD 3, BC 3 and H 3), including a reducible case (A 1 ⊗ 3 ).