You may be right, I think it was a civil health care debate. 
Hummers are known for their aggressiveness and being very territorial (very will documented). My description of the encounter says in fact they were communicating, the black-chinned said get out of here! They were not playing tag as the Anna's came back to feed a few minutes later.
"Fight" is probably a little tough, but certainly these two were behaving like 1 year old kids disagreeing over the ownership of a toy.
Latest Birds: Sage Thrasher, Ferruginous Hawk, Common Goldeneye