  In this paper a quantum mechanics is built by means of a non-Hermitian
momentum operator. We have shown that it is possible to construct two Hermitian
and two non-Hermitian type of Hamiltonians using this momentum operator. We can
construct a generalized supersymmetric quantum mechanics that has a dual based
on these Hamiltonians. In addition, it is shown that the non-Hermitian
Hamiltonians of this theory can be related to Hamiltonians that naturally arise
in the so-called quantum finance.
