The BASE collaboration at the antiproton decelerator/ELENA facility of CERN compares the fundamental properties of protons and antiprotons with ultra-high precision. Using advanced Penning trap systems, we have measured the proton and antiproton magnetic moments with fractional uncertainties of 300 parts in a trillion (p.p.t.) and 1.5 parts in a billion (p.p.b.), respectively. The combined measurements improve the resolution of the previous best test in that sector by more than a factor of 3000. Very recently, we have compared the antiproton/proton charge-to-mass ratios with a fractional precision of 16 p.p.t., which improved the previous best measurement by a factor of 4.3. These results allowed us also to perform a differential matter/antimatter clock comparison test to limits better than $$3\,$$ 3 %. Our measurements enable us to set limits on 22 coefficients of CPT- and Lorentz-violating standard model extensions (SME) and to search for potentially asymmetric interactions between antimatter and dark matter. In this article, we review some of the recent achievements and outline recent progress towards a planned improved measurement of the antiproton magnetic moment with an at least tenfold improved fractional accuracy. Graphic Abstract