In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators
q
^
and
p
^
similar, in a suitable sense, to the self-adjoint position and momentum operators
q
^
0
and
p
^
0
usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be
biorthogonal distributions
, and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with
q
^
and
p
^
, based on the so-called
quasi *-algebras
.