here is the thing, when an alternator fails, it is often the result of bad diodes. the alternator will continue to give out power, just not enough to charge the battery. The voltage of the battery is not a good indication of its quality, since if there is any charge on it at all, it will be 12 volts, unless one or more of the cells are shorted internally. It has six cells, and each cell puts out one volt. But the battery can be bad if the current capacity is low even though the voltage is good. If you buy a new battery, be sure to get a warranty. Or have them put a slow charge on the battery, and re-test it after 24 hours. If it then tests good, replace the alternator.