  We propose a two-dimensional non-Hermitian Chern insulator with inversion
symmetry, which is anisotropic and has staggered gain and loss in both x and y
directions. In this system, conventional bulk-boundary correspondence holds.
The Chern number is a topological invariant that accurately predicts the
topological phase transition and the existence of helical edge states in the
topologically nontrivial gapped phase. In the gapless phase, the band touching
points are isolated and protected by the symmetry. The degenerate points alter
the system topology, and the exceptional points can destroy the existence of
helical edge states. Topologically protected helical edge states exist in the
gapless phase for the system under open boundary condition in one direction,
which are predicted by the winding number associated with the vector field of
average values of Pauli matrices. The winding number also identifies the
detaching points between the edge states and the bulk states in the energy
bands. The non-Hermiticity also supports a topological phase with zero Chern
number, where a pair of in-gap helical edge states exists. Our findings provide
insights into the symmetry protected non-Hermitian topological insulators.
