In the late nineties Bender has started a research program on what is called PT symmetric QM, or non hermitian QM, in which he has shown that if the hamiltonian enjoys a PT symmetry then the spectrum is real even if the hamiltonian was not hermitian. Now a search for "PT symmetric" on the arxiv will return over 400 papers.
This relaxation of the hermiticity of the hamiltonian has led to new forms of hamiltonians that have been rejected before because they were non hermitian. It turns out that such PT symmetric hamiltonians can describe real physical systems.
Since this PT symmetric QM cannot be considered a generalization to conventional quantum mechanics I suppose in which observables are mathematically represented by hermitian operators, I find it intriguing that something like this can describe real physical systems. So how can one understand PT symmetric QM from first principles? I mean how does it fit in the grand picture, why it works?