Friend suspects a bad alternator in an '04 60 hp Merc four-stroke. If he starts the motor and then unhooks a cable at the battery, should the alternator keep the motor running?

The battery was tested by an auto mechanic and tested as good. After a short time of running, the motor will die and the battery will not turn it over. He switched to a TM battery and motor would run for a while and die. Battery would not turn motor over. All batteries were tested as good.

Any other ideas? Thanks.

Joe