I'd like a big cat, preferably panther.

I agree that having a animal companion would be more TES than having humanoid ones, in my view at least. A dog similar to the Alaskan Malamut(or another awesome looking polar dog) would be my companion of choice.
I think that you should be able to buy a dog and name it yourself. If it died you could buy another one(after going through several steps of anger and denial, and finally a funeral, off course). The dog could also be of either gender(more of a cosmetic thing, but still) and maybe even several breeds with different stats. Some dog would fit more with the warrior-type character and some with stealth characters.
Abusing the dog could turn it aggressive towards everyone, but if treated with respect it was more calm, yet more loyal. A more aggressive dog could have increased dmg while a calmer, more loyal one could have more stamina and health(none of these bonuses should make too much difference of course).
In the cities some would be afraid of dogs and aggressive dog might attack while a calm one would not react.
Just some ideas of the top of my head that I think would increase immersion quite a bit.
What do you think?
Edit: making some money breeding dogs would also be kinda cool.

I am the warrior. All I need is a good sword by my side and a good foe to slay.

Also maybe a good tankard of Mead and a bad maid to [censored].

Intimidation companions ftw!
Highwayman: Your money or your life
Me: Dude, wanna see my spider? (show him the tarantula you stole from the spider nest)
Highwayman: Actually, do you accept donations? I have too much money already

An eagle would be pretty cool, but failing that, I suppose a dog.

A wolf or bird would be nice, but personally a big bear would be the best.