  We study the Schr\"odinger operator with a potential given by the sum of the
potentials for harmonic oscillator and imaginary cubic oscillator and we focus
on its pseudospectral properties. A summary of known results about the operator
and its spectrum is provided and the importance of examining its pseudospectrum
as well is emphasized. This is achieved by employing scaling techniques and
treating the operator using semiclassical methods. The existence of
pseudoeigenvalues very far from the spectrum is proven, and as a consequence,
the spectrum of the operator is unstable with respect to small perturbations
and the operator cannot be similar to a self-adjoint operator via a bounded and
boundedly invertible transformation. It is shown that its eigenfunctions form a
complete set in the Hilbert space of square-integrable functions; however, they
do not form a Schauder basis.
