We study the stability of non-ergodic but extended (NEE) phases in
non-Hermitian systems. For this purpose, we generalize a so-called
Rosenzweig-Porter random-matrix ensemble (RP), known to carry a NEE phase along
with the Anderson localized and ergodic ones, to the non-Hermitian case. We
analyze, both analytically and numerically, the spectral and multifractal
properties of the non-Hermitian case. We show that the ergodic and the
localized phases are stable against the non-Hermitian nature of matrix entries.
However, the stability of the fractal phase depends on the choice of the
diagonal elements. For purely real or imaginary diagonal potential the fractal
phases is intact, while for a generic complex diagonal potential the fractal
phase disappears, giving the way to a localized one.
