  We present a non-Hermitian Floquet model with topological edge states in real
and imaginary band gaps. The model utilizes two stacked honeycomb lattices
which can be related via four different types of non-Hermitian time-reversal
symmetry. Implementing the correct time-reversal symmetry provides us with
either two counterpropagating edge states in a real gap, or a single edge state
in an imaginary gap. The counterpropagating edge states allow for either
helical or chiral transport along the lattice perimeter. In stark contrast, we
find that the edge state in the imaginary gap does not propagate. Instead, it
remains spatially localized while its amplitude continuously increases. Our
model is well-suited for realizing these edge states in photonic waveguide
lattices.
