What could be wrong? I've replaced the alternator & battery, but..?

I have a 2004 Honda Civic, whenever I would be stopped at a stop light or something my engine would kind of get louder like it was working harder and then fade back down to normal and my headlights would dim down then go back to regular so I had my alternator checked and was told it needed replaced. I got a brand new alternator put on, I had a brand new battery put in..and it's STILL doing this. Would could be wrong? any ideas? it has 83,XXX miles..