  Moir\'e materials provide a remarkably tunable platform for topological and
strongly correlated quantum phases of matter. Very recently, the first zero
field Abelian fractional Chern insulators (FCIs) have been experimentally
demonstrated and it has been theoretically predicted that non-Abelian states
with Majorana fermion excitations may be realized in the nearly dispersionless
minibands of these systems. Here we provide telltale evidence in terms of
low-energy quantum numbers, spectral flow and entanglement spectra for the even
more exotic possibility of moir\'e-based non-Abelian FCIs exhibiting Fibonacci
parafermion excitations.
