Kaplansky’s unit conjecture states that the only units in the group ring of a torsion-free group should be the “obvious” ones, namely scalar multiples of group elements. Unlike the zero divisor conjecture (the group ring of a torsion-free group is an integral domain), on which steady progress has been made (although it is still not solved in general), the unit conjecture has so far resisted attacks on all but “small” cases. In this talk I will look at a potential minimal counterexample, the so-called Passman fours group, and on recent work with Peter Pappas, where we analyse units for this group.