Photoluminescence of p-type modulation doped (Cd,Mn)Te quantum wells is studied with carrier density up to 5 X 10(11) cm(-2) at various spin splittings. This splitting can be made larger than the characteristic energies of the system thanks to the giant Zeeman effect. At small spin splitting and regardless of the carrier density, the photoluminescence exhibits a single line, which corresponds to the charged exciton in the singlet state. Above a certain spin splitting, the charged exciton is destabilized in favor of the exciton at vanishing hole density, and in favor of a double line at higher carrier density. It is found here that the charged exciton destabilization energy hardly depends on the carrier density. The double line is found to be band-to-band like, with the same initial state - where the holes have the same spin orientation - and final states that differ by some excitation of the 2D hole gas. In addition, the spin splitting needed to fully polarize the hole gas is twice smaller than expected from the single particle image and gives a unique insight into many-body effects in the hole gas.