  We construct complex root spaces remaining invariant under antilinear
involutions related to all Coxeter groups. We provide two alternative
constructions: One is based on deformations of factors of the Coxeter element
and the other based on the deformation of the longest element of the Coxeter
group. Motivated by the fact that non-Hermitian Hamiltonians admitting an
antilinear symmetry may be used to define consistent quantum mechanical systems
with real discrete energy spectra, we subsequently employ our constructions to
formulate deformations of Coxeter models remaining invariant under these
extended Coxeter groups. We provide explicit and generic solutions for the
Schroedinger equation of these models for the eigenenergies and corresponding
wavefunctions. A new feature of these novel models is that when compared with
the undeformed case their solutions are usually no longer singular for an
exchange of an amount of particles less than the dimension of the
representation space of the roots. The simultaneous scattering of all particles
in the model leads to anyonic exchange factors for processes which have no
analogue in the undeformed case.
