Dome C is likely to be the best site in the world for mm and sub-mm observations. In this paper we focus on what can be done from Dome C to investigate the detailed properties of the Cosmic Microwave Background (CMB). Two experiment typologies are particularly promising: precision measurements of the polarization of the CMB, to confirm the presence of an inflation phase in the very early universe, and high resolution measurements of the Sunyaev-Zeldovich effect (SZE) in clusters of galaxies, which can be used to investigate dark energy and dark matter in the Universe. Several important teams are currently carrying out experiments of the first kind; Dome C is the location of the BRAIN experiment, which uses bolometric interferometry as the tool to produce sensitive measurements with low systematic effects, and certainly orthogonal to the systematic effects of all other instruments currently developed to this purpose. Dome C will be an ideal location for a large dish telescope for mm and sub-mm measurements. In addition to cutting-edge sub-mm science, a telescope complementing SPT (in size and/or in frequency) will be ideal for special CMB observation, like the detection of non-Gaussian features, the measurement of relativistic effects in SZE, the measurement of the SZE resulting from the decay products of super-symmetric dark matter in selected clusters.