Not to try to say Im an end-all expert, but i have 2+ years of military, college level training backed by years of military experience with electronics.
A bit more than the guy who took the DeVry class and made a power supply.
Im going to lay out a few FACTS here. Capacitors may do a lot of things, but realistically they only have a few effects. The rest are sort of, like side-effects, in a circuit. Hang with me here, I wont get overly technical.
Here is the major one, I'll bold it for you. Capacitors, OPPOSE a change in voltage, up or down.
How do they do this?
By relasing that stored energy everyone talks about, and charging and drawing power. Because of that everyone seems to think of them as little batteries. This is why that guy's DC power supply steadied out, the cap resisted the voltage jumps.
Charge and Discharge times in a cap are inversely porportional to each other, and are dependent on type/material, and size. (Farads). Charging one of those caps takes a good 30 secs to a minute, right off the battery. So if it discharges just as fast, how is that slow trickle of miniscule power going to aid the amp? You cannot change the discharge time of a cap, and it doesnt under any circumstances like hard bass notes. Caps used in real electronics are on the micro and pico farad scale .000001 and .000000000001 farads, respectively. They charge, and discharge extremely fast.
These gigantic caps, just like any other cap, can add to line voltage, then immediately start to charge exactly as fast as it discharged. Hmm, this is ideal when you have a load(anything that draws power) that stays constant, but you have an unreliable supply.
However, we have the opposite, an unreliable load (power demands constantly varying), with a reliable (12-14.4V) supply. The cap is between the amp and the battery, not the between the battery and the lights. More likely than not, people curing their headlight dimming problem are probably choking their amp somewhat.(Saving headlights from the amp, rather than the other way around) It might not be noticable if you're not already near saturation (full gain or so) on your amp, but these caps are an expensive way to keep your headlights from dimming. The likelyhood also exists that these caps can actually detract from power quality in certain situations, due to their slow response times.
Moral of the story?
There is no magic bullet to increase power output when it simply doesnt exist in the first place.
Higher quality and greater number of batteries will increase the amount power available. Higher power alternator will provide the current to keep said batteries charged in any condition. A flashlight uses 2 AA batteries, not 1 and a capacitor right? Not an exact example, but the cap would discharge so fast when the battery started to die, then draw on it, and make it worse.
But anyway, thats is how you cure dimming lights, and increase the power to your amp.
Oh, there is a component called an inductor, which opposes a change in CURRENT, which might be helpful. However, they are basically a coil of wire and inherently noisy, and like to introduce interference into nearby electronics, you'll never see them on an external component level in a vehicle.