Recommended reading of this interesting and insightful text should
start from Section 4 which offers a transparent description of an
illustrative example of the complex Darboux transformation of the
standard linear harmonic oscillator potential $V(x)$ into a
non-standard complex potential $U(x)$ of eq. (23) generating an
additional, complex energy level $\epsilon$ called, for historical
reasons, `missing state'. This illustration explains what is meant
by ``quasi-isospectrality" of quantum Hamiltonians $p^2+V$ and
$p^2+U$. Then one may proceed to Section 2 which explains how one
can make the latter two Hamiltonians strictly isospectral, getting
rid of the above-mentioned exceptional and uncomfortable complex
`missing level' by a suitable re-specification of a
Hamiltonian-dependent inner product in Hilbert space. By my
opinion, the latter re-specification should have been accompanied
by the citation of its (presumably, original) recommendation, in a
not too different context, by F. G. Scholtz, H. B. Geyer and F. J.
W. Hahne, in Ann. Phys. (NY) 213 (1992) 74.

 MR2171978
Munoz, R. On the generation of non-Hermitian Hamiltonians with
real spectra by a modified Darboux transform. Phys. Lett. A 345
(2005), no. 4-6, 287--292. 81U15 (81Q10)