If your Alternator is only charging at 13.05 volts, then you have a problem. It should charge at a minimum of 14.00 volts, otherwise there is not enough of a gradient so that the energy can flow into the battery and keep it fully charged. At rest, a fully charged battery will show 12.7 to 12.8volts.Charge state can also be verified by a hydrometer test on a lead acid type battery. Not possible to test a sealed Calcium type with a hydrometer for obvious reasons.
Battery is brand new and is not discharging when the car is off. When I got home an hour ago I started the car up, went fine, then eventually managed to get my test probe onto the B+ terminal of the alternator. Again reads 13.08v direct at the alternator's terminal. It's not chucking out enough juice.
the big worry is the guy I bought it from won't get back to me and ignores my calls.