  We study a non-Hermitian Kitaev chain model that contains three sources of
non-Hermiticity: a constant imaginary potential, asymmetry between hopping
amplitudes $t_{\rm R}$ and $t_{\rm L}$ in the right and left directions, and
imbalance in pair potentials $\Delta_{\rm c}$ and $\Delta_{\rm a}$ for pair
creation and annihilation, respectively. We show that bulk--boundary
correspondence holds in this system; two topological invariants defined in bulk
geometry under a modified periodic boundary condition correctly describe the
presence or absence of a pair of boundary zero modes in boundary geometry under
an open boundary condition. One topological invariant characterizes a
topologically nontrivial phase with a line gap and the other characterizes that
with a point gap. The latter appears only in the asymmetric hopping case of
$t_{\rm R} \neq t_{\rm L}$. These two nontrivial phases are essentially
equivalent except for their gap structures. Indeed, the boundary zero modes do
not disappear across the boundary between them. We also show that the boundary
zero modes do not satisfy the Majorana condition if $\Delta_{\rm c} \neq
\Delta_{\rm a}$ and/or $t_{\rm R} \neq t_{\rm L}$.
