  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
$\mu_p$ 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 $\mu_p$. The techniques required for the proton
can be directly applied to measure the antiproton magnetic moment
$\mu_{\bar{p}}$. An improvement in precision of $\mu_{\bar{p}}$ 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.
