Atom interferometers allow the measurement of the acceleration of freely falling atoms with respect to an experimental platform at rest on Earth's surface. Such experiments have been used to test the universality of free fall by comparing the acceleration of the atoms to that of a classical freely falling object. In a recent paper, Müller, Peters and Chu [Nature {\bf 463}, 926-929 (2010)] argued that atom interferometers also provide a very accurate test of the gravitational redshift (or universality of clock rates). Considering the atom as a clock operating at the Compton frequency associated with the rest mass, they claimed that the interferometer measures the gravitational redshift between the atom-clocks in the two paths of the interferometer at different values of gravitational potentials. In the present paper we analyze this claim in the frame of general relativity and of different alternative theories, and conclude that the interpretation of atom interferometers as testing the gravitational redshift at the Compton frequency is unsound. The present work is a summary of our extensive paper [Wolf et al., arXiv:1012.1194, hal-00543979, Class. Quant. Grav. 28, 145017, (2011)], to which the reader is referred for more details.