  Describing open quantum systems in terms of effective non-Hermitian
Hamiltonians gives rise to non-unitary time evolution. In this paper, we study
the impact of non-unitary dynamics on the emergent hydrodynamics in quantum
systems with a global conservation law. To this end, we demonstrate how
linear-response correlation functions can be generalized and interpreted in the
case of non-Hermitian systems. Moreover, we show that dynamical quantum
typicality provides an efficient numerical approach to evaluate such
correlation functions, even though the non-unitary dynamics leads to subtleties
that are absent in the Hermitian case. As a point of reference for our
analysis, we consider the Hermitian spin-$1/2$ XXZ chain, whose
high-temperature transport properties have been characterized extensively in
recent years. Here, we explore the resulting hydrodynamics for different
non-Hermitian perturbations of the XXZ chain. We also discuss the role of
integrability by studying the complex energy-level statistics of the
non-Hermitian quantum models.
