  We describe a novel class of quantum mechanical particle oscillations in both
relativistic and nonrelativistic systems based on $PT$ symmetry and $T^2=-1$,
where $P$ is parity and $T$ is time reversal. The Hamiltonians are chosen at
the outset to be self-adjoint with respect to a PT inner product. The quantum
mechanical time evolution is based on a modified CPT inner product constructed
in terms of a suitable C operator. The resulting quantum mechanical evolution
is shown to be unitary and probability is conserved by the oscillations.
