  We introduce, for the first time, bicoherent-state path integration as a
method for quantizing non-hermitian systems. Bicoherent-state path integrals
arise as a natural generalization of ordinary coherent-state path integrals,
familiar from hermitian quantum physics. We do all this by working out a
concrete example, namely, computation of the propagator of a certain
quasi-hermitian variant of Swanson's model, which is not invariant under
conventional $PT$-transformation. The resulting propagator coincides with that
of the propagator of the standard harmonic oscillator, which is isospectral
with the model under consideration by virtue of a similarity transformation
relating the corresponding hamiltonians. We also compute the propagator of this
model in position space by means of Feynman path integration and verify the
consistency of the two results.
