Sudden drop in battery voltage

you only register 12 volts with a new battery?? Someone lied to you about the battery being new when you bought it. it should be a good pinch higher than that although the alt is fine. you could have a low acid level. Take top off and look and see if you have plenty acid in your battery. Your cells should be covered, since you say it's new. interesting story

Thanks for your answer Randy. The new battery must be ok as I use it to jump start the car with.

The battery should be about 12.6 volts with the engine off. The alternator puts out a higher voltage to charge the battery. 14.6 volts just after starting the car is normal. It should drop down to around 13.5 after a few minutes.

I think you have poor battery connections. Clean and tighten the connections.