  We investigate the many-particle and mean-field correspondence for a
non-Hermitian N-particle Bose-Hubbard dimer where a complex onsite energy
describes an effective decay from one of the modes. Recently a generalized
mean-field approximation for this non-Hermitian many-particle system yielding
an alternative complex nonlinear Schr\"odinger equation was introduced. Here we
give details of this mean-field approximation and show that the resulting
dynamics can be expressed in a generalized canonical form that includes a
metric gradient flow. The interplay of nonlinearity and non-Hermiticity
introduces a qualitatively new behavior to the mean-field dynamics: The
presence of the non-Hermiticity promotes the self-trapping transition, while
damping the self-trapping oscillations, and the nonlinearity introduces a
strong sensitivity to the initial conditions in the decay of the normalization.
Here we present a complete characterization of the mean-field dynamics and the
fixed point structure. We also investigate the full many-particle dynamics,
which shows a rich variety of breakdown and revival as well as tunneling
phenomena on top of the mean-field structure.
