  Normalizing flows have recently been applied to the problem of accelerating
Markov chains in lattice field theory. We propose a generalization of
normalizing flows that allows them to applied to theories with a sign problem.
These complex normalizing flows are closely related to contour deformations
(i.e. the generalized Lefschetz thimble method), which been applied to sign
problems in the past. We discuss the question of the existence of normalizing
flows: they do not exist in the most general case, but we argue that exact
normalizing flows are likely to exist for many physically interesting problems,
including cases where the Lefschetz thimble decomposition has an intractable
sign problem. Finally, normalizing flows can be constructed in perturbation
theory. We give numerical results on their effectiveness across a range of
couplings for the Schwinger-Keldysh sign problem associated to a real scalar
field in $0+1$ dimensions.
