  Identifying and understanding interacting systems that can host non-Abelian
topological phases with fractionalized quasiparticles have attracted intense
attentions in the past twenty years. Theoretically, it is possible to realize a
rich variety of such states by coupling two Abelian fractional quantum Hall
(FQH) states together through gapping out part of the low energy degrees of
freedom. So far, there are some indications, but no robust example has been
established in bilayer systems for realizing the non-Abelian state in the past.
Here, we present a phase diagram of a double-layer bosonic FQH system based on
the exact diagonalization and density-matrix renormalization group (DMRG)
calculations, which demonstrate a potential regime with the emergence of the
non-Abelian bosonic Moore-Read state. We start from the Abelian phase with
fourfold topological degeneracies on torus geometry when the two layers are
weakly coupled. With the increase of interlayer tunneling, we find an
intermediate regime with a threefold groundstate degeneracy and a finite
fractional drag Hall conductance. We find the different topological sectors in
consistent with Moore-Read state by inserting different fluxes in adiabatic
DMRG study. We also extract the modular $\mathcal{S}-$ matrix, which supports
the emergence of the non-Abelian Ising anyon quasiparticle in this system.
