Recent exciting progress in the preparation and manipulation of the motional quantum states of a single trapped proton enabled the first direct detection of the particle's spin state. Based on this success the proton magnetic moment μp was measured with ppm precision in a Penning trap with a superimposed magnetic field inhomogeneity. An improvement by an additional factor of 1000 in precision is possible by application of the so-called double Penning trap technique. In a recent paper we reported the first demonstration of this method with a single trapped proton, which is a major step towards the first direct high-precision measurement of μp. The techniques required for the proton can be directly applied to measure the antiproton magnetic moment μp. An improvement in precision of μp by more than three orders of magnitude becomes possible, which will provide one of the most sensitive tests of CPT invariance. To achieve this research goal we are currently setting up the Baryon Antibaryon Symmetry Experiment (BASE) at the antiproton decelerator (AD) of CERN.