This video isn't here on youtube with any (good) english tags so I've re-uploaded it.
This is a TAS (tool assisted) battle probably between two (or maybe someone doing it by himself) human players. Needless to say it's better than any videos of people playing in real time, but it's the most impressive TAS battle I've seen to date.
Just a little info if you're not getting it - when you do a huge combo, you 'send lines' over to your opponent, but stuff doesn't start dropping until your combo finishes. So while your combo is going on, your opponent is building up his own combo to counter. So when your combo finishes, he starts his, which first cancels any damage you sent over to his side, and then starts sending damage over to your side. And so it continues...
Nico nico version:
http://www.nicovideo.jp/watch/sm7597646
I don't know much about this game so if I got something wrong or you have something to add, just leave a comment.