  Non-Hermitian physics in open quantum many-body systems provides novel
opportunities for the discovery of exotic quantum phenomena unexpected in
Hermitian systems. A previous study of the non-Hermitian Kondo problem in
ultracold atoms reports reversion of renormalization group flows which violates
the $g$ theorem and produces an unusual quantum phase transition. In this work,
we study the effect of electron-electron interactions by considering the
non-Hermitian Kondo problem in a Luttinger liquid. By performing a perturbative
renormalization group analysis to two-loop order, we find that the interplay
between non-Hermitian Kondo couplings and electron-electron interactions can
produce a pair of complex fixed points. Complex fixed points have often been
discussed in an attempt to understand the extremely long correlation length of
Hermitian systems with weakly first-order transitions. Here, we show that
complex fixed points arise naturally and can be physically realized in open
quantum systems. We discuss consequences of the complex fixed points and future
directions.
