I would love to see pets have emotional responses, and related non-combat utility functions.
Examples of emotional responses: acting sad when you die.
Acting angry or aggressive towards a toon or person you are attacking.
Acting happy when you defeat a boss.

There could also be related companion utility switches (so you can turn them on and off).

For example, a switch to automatically fetch loot within a short distance.
That wouldn't change the game balance, but word add a fun factor to it.
Other possible pet switches...
Artifact tracking...
Have pets jump and point at nearby artifacts.
Quest tracker.
Pets lead you to the closest missing quest, or quest objective in a zone. They could make an appropriate gesture when the quest / needed baddie / item is close (angry, excited jumping, digging, acting sad, etc).
Carnage pointer.
Pets point you towards missing carnage in a zone. Acts aggressive when the baddie is close.

Basically, make them an emotional and useful companion instead of a cute walking statue.