This is definitely a common thing among proteges. It happened in my mazdaspeed and it happens (seemingly worse) in my protege5...
when I use the windows my headlights used to dim (before I got HIDs) and the interior gauge display dims too. Now when I turn on my HIDs my sound system shuts off for a few seconds and turns back on.. but my headlights don't dim anymore because after the initial startup, HIDs use less power.
the 1200 watt sound system in my car probably isn't helping my case (everything dims when I am at full volume that has a light lol)
However, there are 1 of 2 options I would suggest, both have already been mentioned. Since I am a car audio buff, the first one I am going to recommend is "the big 3"
Get yourself some 4 gauge wire (0 if your feeling it) and follow one of the links above. This will allow more free flowing current and should help a little. I always recommend to do this first
If that doesn't work, dont buy a replacement oem alternator.. get yourself a high output alternator. I think our car is only 70amp stock which is incredibly low, but you can get a 140 amp for the p5 for $200 online. This should fix everything else. However, if you do the alternator without the big 3 upgrade, it won't be as beneficial as it could be because of the crappy grounding and wiring that is stock
hope my 2 cents helped