Car running there should be no voltage between the negative battery terminal and the alternator shell. If no voltage then the new alternator is starting to sound bad. You would use a clip on wire to stay away from the running engine components on the front of the engine for a ground voltage test. Always practice safely. Usually if the starter will crank the alternator is grounded though. Still not always.

Where did the replacement alternator come from? Mexican rebuilt? Using a jumper cable as already mentioned is also a good ground test.

Reaching down with the engine off you should not be able to turn the alternator fan with your fingers either.