Old fashioned, mechanical gauges moved when the water temperature changed (like, when the thermostat opens, or when pulling up a long hill, or coasting down a long hill), and the alternator / V gauge would move up/down when devices were turned on/off.
Customers complained about these normal variations, so OEMs (since the '70s or '80s) have put buffers into their operation making them nearly useless for much, other than as "idiot lights" (they don't move until something severe has already happened). It is amazing to drive a really old vehicle with mechanical gauges, to see how they move around communicating lots of subtle things to an attentive driver.
On a modern car, if the V gauge is moving around, that does seem like it would be wise to test the system to confirm what is actually going on. Most shops will provide a free charging system test, or for a very nominal fee.
Can you define exactly what "moves around" means?
How much is the needle moving, and under what conditions?
Drops low at idle, and moves higher as RPM increases?
Or swings wildly as the vehicle drives over bumps?
Did you replace the alternator with a "new" OEM alternator, or a cheap auto parts store rebuild?
How old is the battery?
Slight needle movement as electrical loads are switched on and off, or a slight difference between voltage at idle and cruise speed is normal. Larger, sudden swings in needle position is not.