  Using as a prototype example the harmonic oscillator we show how losing
self-adjointness of the hamiltonian $H$ changes drastically the related
functional structure. In particular, we show that even a small deviation from
strict self-adjointness of $H$ produces two deep consequences, not well
understood in the literature: first of all, the original orthonormal basis of
$H$ splits into two families of biorthogonal vectors. These two families are
complete but, contrarily to what often claimed for similar systems, none of
them is a basis for the Hilbert space $\Hil$. Secondly, the so-called metric
operator is unbounded, as well as its inverse.
  In the second part of the paper, after an extension of some previous results
on the so-called $\D$ pseudo-bosons, we discuss some aspects of our extended
harmonic oscillator from this different point of view.
