Ideally you want both a second battery and a high output alt. The alt makes the power, the batteries are there to stabilize it. A single battery can only handle so much current draw before the float voltage (whatever voltage the alt is charging at) is drawn down to the nominal voltage (12 V thereabouts) It's very little strain on the alt to put the battery back to float voltage unless the battery has been pretty deeply discharged. With the car running and playing music, not test tones, this will pretty much never happen usually even with just the stock alt, since the current draw with music does not average out to a very high draw and is nowhere near continuous.
If you have ever used an air compressor, it really is a good analogy. The compressor itself is pumps the air into the pressure tank just like the alt keeps the batteries charged. When you demand air from the system, the tank provides the initial hit of air and then the compressor kicks in to maintain the pressure for long term demand and to get ready for the next big demand. The alt and the batteries work in a very similar fashion. The alt provides continuous power to the system, but it does not react well to rapidly changing current demands, the voltage goes all over the place. The batteries act as a buffer to take the bit current hits and keep the alt from having to try to rapidly change output. The overall demand on the alt is actually reduced by having a good buffer in place. A really huge cap (100 or so Farads) would do this if the high ESR of a cap didn't drop the voltage below a usable level before it even leaves the cap.