I'd really, really like to see Hulk display the two parts of himself:
1. I'd like to see him struggling, but winning the internal war to control himself in non-combat and relatively stress-free situations BUT still use more of his baser abilities/instincts/attributes.
-For example, (and this is gonna sound weird until you really think about it), I've always thought Hulk should SNIFF a lot of things, especially when he is introduced to someone new to him. The sense of smell is one of the most powerful memory triggers human beings have, and it would show Banner, somewhere deep in the recesses of the Hulk's mind, trying to bring this monster to some sort of understanding, especially with people/things he encountered as Banner, and is now encountering as Hulk.
I could see Hulk encountering Black Widow for the first time, AFTER Banner has encountered her. Hulk walks up to her, and lifts a lock of her hair, and sniffs it, and only THEN does he realize he has encountered her before...
Additionally, we have all seen dogs show their dissaproval or disdain for a person by sniffing them, then doing that quasi "sneeze/exhale" thing they do to rid their nostrils of an odor they don't like. Gorillas do the same thing. I could see Hulk walking up to Thor, sniffing at him, then doing the sneeze/exhale thing to show his disdain for Thor, especially if it is after a tense confrontation.
These time could also show Banner rising to the surface of the Hulk's consciousness, by Hulk being able to answer simple questions and carry on a very fundamental conversation.
HOWEVER...
2. When stressed out, I want to see the more animalistic side of Hulk come out, and see/hear nothing out of him but the screams, roars, and grunts from the last movies...