How does non-Hermitian quantum mechanics (PT-symmetric QM) fit in physics? 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? 
 A: Quoting Ali Mostafazadeh in arXiv:quant-ph/0307059:

It can also be shown that whenever $H$ is a diagonalizable operator with a real spectrum,
  then it can be mapped to a Hermitian operator via a similarity transformation.

The $PT$-symmetric operators are of this type.
Though I haven't read it (yet), arXiv:0810.5643 by the same author should answer your questions in detail.
Update: Skimming over the article, I found the following remark:

The main disadvantage of employing the Hermitian representation is that in
  general the Hamiltonian $h$ is a terribly complicated nonlocal operator. Therefore, the computation of the energy levels and the description of the dynamics are more conveniently carried out in the pseudo-Hermitian representation. In contrast, it is the Hermitian representation that facilitates the computation of the expectation values of the physical
  position and momentum operators as well as that of the localized states in physical position or momentum spaces.

Note that the Klein-Gordon field is probably a good example for this, whose treatment in terms of classical quantum mechanics is the subject of the first paper I linked.
A: It is the formulation of Hamiltonian QM in a nontraditional inner product, in which certain (in the standard inner product) nonhermitian Hamiltonians become
selfadjoint.
Calling it a generalization of QM is not appropriate. It has very little impact on most of quantum mechnaics, and is nothing of general interest, just a theoretical playground for afficionados.
A: To address your question and to add a little bit of extra flavour to Christoph's answer I think something like this text is a good starting point: Non-Hermitian Quantum Mechanics by Nimrod Moiseyev
I think some of the answers to this question are also missing a factor which this book and other similar texts address. 
From the perspective of condensed matter, non-Hermicity can be viewed as consequence of mathematically describing an open system, or certain representations of specific bosonic systems. PT-symmetric Hamiltonians are simply a special case of this and in some examples (of those 400 arxiv results you refer to) contain descriptions of systems where the most important consequence is the non-conservation of some operator (and hence the non-unitarity). Specifically thinking about time evolution helps here. If one considers splitting a Hamiltonian into it's Hermitian and Non-Hermitian parts, there is clearly no Hermitian counterpart to the additional phases picked up by non-Hermitian contributions when encircling degeneracies. This has been shown in photonic systems with absorptive (source) and amplifying (sink) like components, which produce phenomena that cannot be described in terms of Hermitian operators/physics. (i.e consider the following two papers: arxiv:1508.03985, arXiv:1603.05312) You can find a good dozen examples of this on the arXiv. 
You're absolutely right to think that PT symmetric QM isn't a generalisation of QM, it's complementary in it's own right.       
A: My experience is based on pseudo hermitian hamiltonians, which are reducible to
an hermitian one, but they end up to be complicated hamiltonians that are generally not
interpretable from a physics point of view. 
Hermitian hamiltonians are more physical
in the sense of reciprocity of the interaction; the interaction of 2 on 1 is also
the interaction of 1 on 2.
The mathematics of pseudo hermitian quantum physics is not new at all; it reduces
to a Sturm Liouville formalism that is a natural generalization of Schrodinger.
Therefore I think that workers in this field have quite a struggle ahead of them
to make any of this really usefull or novel.
