  A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum
system, if we can modify the Hilbert space of state vectors of the system so
that it turns into a Hermitian operator. If this operator is time-dependent,
the modified Hilbert space is generally time-dependent. This in turn leads to a
generic conflict between the condition that the Hamiltonian is an observable of
the system and that it generates a unitary time-evolution via the standard
Schr\"odinger equation. We propose a geometric framework for addressing this
problem. In particular we show that the Hamiltonian operator consists of a
geometric part, which is determined by a metric-compatible connection on an
underlying Hermitian vector bundle, and a non-geometric part which we identify
with the energy observable. The same quantum system can be locally described
using a time-dependent Hamiltonian that acts in a time-independent state space
and is the sum of a geometric part and the energy operator. The full global
description of the system is achieved within the framework of a moderate
geometric extension of quantum mechanics where the role of the Hilbert space of
state vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed with
a metric compatible connection, and observables are given by global sections of
a real vector bundle that is determined by $\mathcal{E}$. We examine the
utility of our proposal to describe a class of two-level systems where
$\mathcal{E}$ is a Hermitian vector bundle over a two-dimensional sphere.
