  Recent experimental observations of the fractional quantum anomalous Hall
effect in spin/valley polarized moir\'{e} systems call for a more expansive
theoretical exploration of strongly correlated physics in partially filled
topological bands. In this work we study a state that we refer to as the
conjugate-composite Fermi liquid (cCFL), which arises when a pair of Chern
bands with opposite Chern numbers are both half-filled. We demonstrate that the
cCFL is the parent state of various interesting phenomena. As an example, we
demonstrate that with the existence of an inplane spin order, the cCFL could be
driven into a quantum bad metal phase, in the sense that it is a metallic phase
whose zero temperature longitudinal resistivity is finite, but far greater than
the Mott-Ioffe-Regal limit, i.e. $\rho^e_{xx} \gg h/e^2$. The bad metal phase
is also accompanied with a new Wiedemann-Franz law, meaning the thermal
conductivity is proportional to the electrical resistivity rather than
conductivity. Other proximate phases of the cCFL such as superconductivity and
a chiral spin liquid phase can occur when the composite fermions (CF) form the
inter-valley CF-exciton condensate.
