  We propose a pair of the complex Berry curvatures associated with the
non-Hermitian Hamiltonian and its Hermitian adjoint to reveal new physics in
non-Hermitian systems. We give the complex Berry curvature and Berry phase for
the two-dimensional non-Hermitian Dirac model. The imaginary part of the
complex Berry phase induces the quantum Hall susceptance such that the quantum
Hall conductance is generalized to quantum Hall admittance for non-Hermitian
systems, which implies that the non-Hermiticity of systems could induce an
intrinsic quantum capacitance or inductance of systems depending on the
non-Hermitian parameters. We analyze the complex energy band structures of the
two-dimensional non-Hermitian Dirac model and demonstrate the point and line
gaps and their closings as the exceptional degeneracy of the energy bands in
the parameter space, which are associated with topological phases. In the
continuum limit, we obtain the complex Berry phase in the parameter space.
