  By constructing germanene (silicene, stanene)/$M$Bi$_2$Te$_4$ ($M$ =
3d-transition elements) heterostructures, we discovered and designed
multimechanism quantum-anomalous-Hall (QAH) systems, including $\Gamma$-based
QAH, $K$-$K'$-connected QAH, and valley-polarized $K$- or $K'$-based QAH states
via first-principle computations. The unique systems possess a global gap and
tunable Chern number. The coexisting conventional $\Gamma$-based QAH state of
$M$Bi$_2$Te$_4$ and valley-polarized $K$($K'$)-based QAH state of germanene
(silicene, stanene), with opposite chirality, can interact with each other.
Adjusting magnetic configurations of $M$Bi$_2$Te$_4$-layers not only switch on
(off) the QAH conductance, but also modulate Chern numbers exactly. For
example, the germanene/bilayer-NiBi$_2$Te$_4$ possesses the Chern number $C =
+1$ in ferromagnetic couplings and $C = +2$ in antiferromagnetic couplings. The
novel multimechanism QAH insulators, which are achievable in experiments,
provide a new approach to spintronics and valleytronics based on topological
states of matter.
