Tracing the formation and evolution of all supermassive black holes, including the obscured ones, understanding how black holes influence their surroundings and how matter behaves under extreme conditions, are recognized as key science objectives to be addressed by the next generation of instruments. These are the main goals of the COSPIX proposal, made to ESA in December 2010 in the context of its call for selection of the M3 mission. In addition, COSPIX, will also provide key measurements on the non thermal Universe, particularly in relation to the question of the acceleration of particles, as well as on many other fundamental questions as for example the energetic particle content of clusters of galaxies. COSPIX is proposed as an observatory operating from 0.3 to more than 100 keV. The payload features a single long focal length focusing telescope offering an effective area close to ten times larger than any scheduled focusing mission at 30 keV, an angular resolution better than 20 arcseconds in hard X-rays, and polarimetric capabilities within the same focal plane instrumentation. In this paper, we describe the science objectives of the mission, its baseline design, and its performances, as proposed to ESA
