  A non-Hermitian operator $H$ defined in a Hilbert space with inner product
$\langle\cdot|\cdot\rangle$ may serve as the Hamiltonian for a unitary quantum
system, if it is $\eta$-pseudo-Hermitian for a metric operator
(positive-definite automorphism) $\eta$. The latter defines the inner product
$\langle \cdot|\eta\cdot\rangle $ of the physical Hilbert space $\mathscr{H}
_{\eta}$ of the system. For situations where some of the eigenstates of $H$
depend on time, $\eta$ becomes time-dependent. Therefore the system has a
non-stationary Hilbert space. Such quantum systems, which are also encountered
in the study of quantum mechanics in cosmological backgrounds, suffer from a
conflict between the unitarity of time evolution and the unobservability of the
Hamiltonian. Their proper treatment requires a geometric framework which
clarifies the notion of the energy observable and leads to a geometric
extension of quantum mechanics (GEQM). We provide a general introduction to the
subject, review some of the recent developments, offer a straightforward
description of the Heisenberg-picture formulation of the dynamics for quantum
systems having a time-dependent Hilbert space, and outline the
Heisenberg-picture formulation of dynamics in GEQM.
