  The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian
$\cal{PT}$-symmetric part, can be transformed into a Hermitian Hamiltonian.
This is achieved by introducing a metric which, in general, renders other
observables such as the usual momentum or position as non-Hermitian operators.
The metric depends on one real parameter, the full range of which is
investigated. The explicit functional dependence of the metric and each
associated Hamiltonian is given. A specific choice of this parameter determines
a specific combination of position and momentum as being an observable; this
can be in particular either standard position or momentum, but not both
simultaneously. Singularities of the metric are explored and their removability
is investigated. The physical significance of these findings is discussed
