  We propose a phase-biased non-Hermitian Josephson junction (NHJJ) composed of
two superconductors mediated by a short non-Hermitian link. Such a NHJJ is
described by an effective non-Hermitian Hamiltonian derived based on the
Lindblad formalism in the weak coupling regime. By solving the Bogoliubov-de
Gennes equation, we find that its Andreev spectrum as a function of phase
difference exhibits Josephson gaps, i.e., finite phase windows with no Andreev
(quasi)bound states. The complex Andreev spectrum and the presence of Josephson
gaps constitute particular spectral features of the NHJJ. Moreover, we propose
complex supercurrents arising from inelastic Cooper pair tunneling to
characterize the anomalous transport in the NHJJ. Additional numerical
simulations complement our analytical predictions. We demonstrate that the
Josephson effect is strongly affected by non-Hermitian physics.
