  We demonstrate the emergence of a topological ordered phase for non-Hermitian
systems. Specifically, we elucidate that systems with non-Hermitian two-body
interactions show a fractional quantum Hall (FQH) state. The non-Hermitian
Hamiltonian is considered to be relevant to cold atoms with dissipation. We
conclude the emergence of the non-Hermitian FQH state by the presence of the
topological degeneracy and by the many-body Chern number for the ground state
multiplet showing $C_{\mathrm{tot}}=1$. The robust topological degeneracy
against non-Hermiticity arises from the many-body translational symmetry.
Furthermore, we discover that the FQH state emerges without any repulsive
interactions, which is attributed to a phenomenon reminiscent of the continuous
quantum Zeno effect.
