  A set of Hamiltonians that are not self-adjoint but have the spectrum of the
harmonic oscillator is studied. The eigenvectors of these operators and those
of their Hermitian conjugates form a bi-orthogonal system that provides a
mathematical procedure to satisfy the superposition principle. In this form the
non-Hermitian oscillators can be studied in much the same way as in the
Hermitian approaches. Two different nonlinear algebras generated by properly
constructed ladder operators are found and the corresponding generalized
coherent states are obtained. The non-Hermitian oscillators can be steered to
the conventional one by the appropriate selection of parameters. In such limit,
the generators of the nonlinear algebras converge to generalized ladder
operators that would represent either intensity-dependent interactions or
multi-photon processes if the oscillator is associated with single mode photon
fields in nonlinear media.
