  Emergence of multiple topological phases with a series of Chern numbers, $\pm
1$, $\mp 1$, $\pm 2$, $\mp 2$, $\pm 3$ and $\mp 4$, are observed in a
ferromagnetic Kitaev-Heisenberg-spin-anisotropic model on honeycomb lattice
with further neighbor interactions in the presence of an external magnetic
field. Magnon Chern insulating dispersions of this two-band model are studied
by using linear spin-wave theory formulated on the exact ferromagnetic ground
state. Magnon edge states are obtained for the respective topological phases
along with density of states. A topological phase diagram of this model is
presented. Behavior of thermal Hall conductivity for those phases is studied.
Sharp jumps of thermal hall conductivity is noted near the vicinity of phase
transition points. Topological phases of the Kitaev spin-liquid compounds,
$\alpha$-RuCl$_3$, X$_2$IrO$_3$, X = (Na, Li) and CrY$_3$, Y = (Cl, Br, I) are
characterized based on this theoretical findings.
