Do I need a higher output battery if I get a higher output alternator?

No,you don't.

If you're running the volume high enough to achieve the full 1,200 watts RMS, and running it like that for extended amounts of time, then the bigger alt. is a great idea. At 1,200 watts RMS, you're asking for at least 105 amperes from your alternator. That's if you're using a class D monoblock amp. If you're using a Class AB, then the amp will need more amperes as they're less efficient than a Class D. Even the cheapest of car batteries have at least 400 cold crank amps, so your battery isn't the problem. The only time you need a higher capacity battery is if you plan on playing your stereo with the car off for extended amounts of time. This is a break down of how it works. The alternator is what supplies the electricity for the car. The battery is a voltage storage device. It also serves as a buffer for the electronics in your car. Sometimes that's bad. As it is being charged it can put additional strain on the alternator while you're enjoying the awesome stereo in your car. If the alternator is only putting out 80 amperes, and the stereo is using 104 amperes, then the battery has now gone from being charged and storing the electricity to supplying electricity. It's not designed for that. It's designed for a large quick supply of power for the starter. That's it. Plus, the alternator is really struggling while trying to keep the battery charged and supply elect. for the stereo, and the ECM, PCM, headlights, etc. Everything is being worn out faster. The battery is never fully being charged, and the alternator is like a fat guy running a 5 mile marathon while holding a beer in one hand and a sandwich in the other. So, for the quick answer... No. You don't need a larger capacity battery unless you plan on playing the stereo without the car running.

If you got a higher output alternator, and need ekstra push to big stereo system, i see no problem doing that. Why in the first place by a big ALT if not buy big battery :) Just make sure bigger cables than whats usually in cars for ground power etc. As in do the big 3 upgrade.

Swap to a high output OPTIMA gel battery