  Inspired by current research on measurement-induced quantum phase
transitions, we analyze the nonunitary Floquet transverse-field Ising model
with complex nearest-neighbor couplings and complex transverse fields. Unlike
its unitary counterpart, the model shows a number of steady phases, stable to
integrability breaking perturbations. Some phases have robust edge modes and/or
spatiotemporal long-range orders in the bulk. The transitions between the
phases have extensive entanglement entropy, whose scaling with the system size
depends on the number of the real quasiparticle modes in the spectrum at the
transition. In particular, the volume law scaling appears on some critical
lines, protected by pseudo-Hermiticity. Both the scaling of entanglement
entropy in steady states and the evolution after a quench are compatible with
the non-Hermitian generalization of the quasiparticle picture of Calabrese and
Cardy at least qualitatively.
