  We present ALMA [CII] 158 $\mu$m line and underlying far-infrared continuum
emission observations ($0''.57 \times 0''.46$ resolution) toward a
quasar-quasar pair system recently discovered at $z = 6.05$ (Matsuoka et al.
2024). The quasar nuclei (C1 and C2) are very faint ($M_{\rm 1450} \gtrsim -23$
mag), but we detect very bright [CII] emission bridging the 12 kpc between the
two objects and extending beyond them (total luminosity $L_{\rm [CII]} \simeq 6
\times 10^9~L_\odot$). The total star formation rate of the system is $\sim
100$ to 550 $M_\odot$ yr$^{-1}$, with a [CII]-based total gas mass of $\sim
10^{11}~M_\odot$. The dynamical masses of the two galaxies are large ($\sim 9
\times 10^{10}~M_\odot$ for C1 and $\sim 5 \times 10^{10}~M_\odot$ for C2).
There is a smooth velocity gradient in [CII], indicating that these quasars are
a tidally interacting system. We identified a dynamically distinct, fast [CII]
component around C1. Detailed inspection of the line spectrum reveals the
presence of a broad wing component, which we interpret as the indication of
fast outflows with a velocity of $\sim 600$ km s$^{-1}$. The expected mass
loading factor of the outflows, after accounting for multiphase gas, would be
$\gtrsim 3-8$, as expected for an AGN-driven outflow. Hydrodynamic simulations
in the literature predicted that this pair will evolve to a luminous ($M_{\rm
1450} \lesssim -26$ mag), starbursting ($\gtrsim 1000~M_\odot$ yr$^{-1}$)
quasar after coalescence, one of the most extreme populations in the early
universe.
