  A scenario of non-Hermitian bulk--boundary correspondence proposed for
one-dimensional topological insulators is adapted to a non-Hermitian Chern
insulator to examine its applicability to two-dimensional systems. This
scenario employs bulk geometry under a modified periodic boundary condition and
boundary geometry under an open boundary condition. The bulk geometry is used
to define a topological number, whereas the boundary geometry is used to
observe the presence or absence of a topological boundary state. It is
demonstrated that the bulk--boundary correspondence holds in a two-dimensional
Chern insulator with gain/loss-type non-Hermiticity; a nontrivial Chern number
calculated in the bulk geometry is in one-to-one correspondence with the
presence of a topological boundary state in the boundary geometry. This
approach enables us to determine a phase diagram in the boundary geometry.
