  Two discrete N-level alternatives to the popular imaginary cubic oscillator
are proposed and studied. In a certain domain ${\cal D}$ of parameters $a$ and
$z$ of the model, the spectrum of energies is shown real (i.e., potentially,
observable) and the unitarity of the evolution is shown mediated by the
construction of a (non-unique) physical, ad hoc Hilbert space endowed with a
nontrivial, Hamiltonian-dependent inner-product metric $\Theta$. Beyond ${\cal
D}$ the complex-energy curves are shown to form a "Fibonacci-numbered"
geometric pattern and/or a "topologically complete" set of spectral loci. The
dynamics-determining construction of the set of the eligible metrics is shown
tractable by a combination of the computer-assisted algebra with the
perturbation and extrapolation techniques. Confirming the expectation that for
the local potentials the effect of the metric cannot be short-ranged.
