Since we seem to have this argument on what appears to be a regular basis let me try and clear a few things up...
Capacitors that are used to try and maintain voltage are relatively useless. The reason behind this is that their storage capacity is so small that when a long bass line comes in and the capacitor discharges to make up for the voltage drop, it has to recharge while the bass line is still playing because the discharge rate is so quick. This, in turn, causes even greater drop because now the capacitor and the amplifier are pulling current. The only use for a capacitor (besides the obvious use of being on a circuit board) is when you need a quick discharge of energy, they are not good for long periods of time because they simply don't have the reserve.
Batteries make up for the poor reserve size of capacitors and are why people favor them over capacitors. However the argument is that even with a battery you will still get voltage drop. And that's true, because a battery is not a fix for voltage drop. A battery, if anything, is just a bandaid fix to make up for the fact that your alternator can't keep up. So if your voltage is dropping way down, all adding a battery will do is provide an extra power source for when your alternator can't charge. This still causes voltage drop, because the battery does not rest at the alternator's charging voltage, it rests at around 12.8. So if your voltage was dropping down to 12.0v, it may only drop down to 12.8 because of the extra battery. If anything, this is good because it prevents too low of a voltage. You may not be able to fight voltage drop with a battery, but you can at least attempt to sustain it at a reasonable level. But then the problem you run into is your alternator was not designed for having to charge an extra battery, so you put a lot more stress onto it. Depending on the conditions (such as typical listening volume, or current draw), the extra stress isn't usually too great, so we can get away with that.
Lastly, obviously you have the alternator which doesn't really have a downside to it besides price I guess. But basically, every argument ends in the fact that a new alternator is the best choice, which you can't argue as being false. However, taking into consideration the conditions of most peoples systems, a new alternator simply isn't necessary. The OP is running a 1400w amp, which on a 12v system is drawing roughly 110A of current. His vehicle most likely has a 80-90A alt, so to dish out the hundreds of dollars for the extra couple dozen amps of available current output just really isn't that needed. It would solve the problems he's experiencing, but realistically speaking, if you were in his shoes I highly doubt you would drop the $ on a new alt when a battery would more than likely take care of the problems he is having.
Moral of the story: Using a cap to prevent dimming is an improper use for it, batteries won't prevent voltage drop but can sustain voltage at a reasonable level, and an alt would be the ideal thing to upgrade, but in most cases it is an unneeded expense.
/poast