  We introduce non-Hermitian generalizations of Majorana zero modes (MZMs)
which appear in the topological phase of a weakly dissipative Kitaev chain
coupled to a Markovian bath. Notably, the presence of MZMs ensures that the
steady state in the absence of decoherence events is two-fold degenerate.
Within a stochastic wavefunction approach, the effective Hamiltonian governing
the coherent, non-unitary dynamics retains BDI classification of the closed
limit, but belongs to one of four non-Hermitian "flavors" of the ten-fold way.
We argue for the stability of MZMs due to a generalization of particle-hole
symmetry, and uncover the resulting topological phase diagram. Qualitative
features of our study generalize to two-dimensional chiral superconductors. The
dissipative superconducting chain can be mapped to an Ising model in a complex
transverse field, and we discuss potential signatures of the degeneracy.
