Viking,
No, you cannot. As CM points out, you can only be co-pilot of an a/c requiring multi-crew. Unlikely, you'll get to fly one of these as a PPL.
However, the role of PIC can change during the flight. I.e. you can be PIC for half the time and the other pilot for the other half (assuming, of course, you're both current and hold valid medicals). In this scenario, a one-hour flight could be logged as 30 mins PIC each.
HTH