Just a basic example from Obivion: You are tasked to retrieve the sword Umbra and return it to Clavicus Vile. Instead of returning the sword, you keep it for yourself.

I, personally, would like to see a little bit of this in ESO. Lets just say a nobleman pays you to collect his family heirloom, but does not tell you what it is, only that you need to pick it up and bring it to him. However, this heirloom is actually a chestplate made of mithril and dragon bones.
You think to yourself, "That nobleman will pay me 4000 gold to bring this to him. Although, this armour DOES look good on me..."

You keep the armour and all is well. Merrily prancing about in your new protection, when after some time, "We've been looking for you. You were one hard nut to track down, but we found you. The nobleman is paying well for that armour - and your head!"
A rather large bounty has been placed on you. 'To he who brings me my prized heirloom and the head of who wears it, the reward of 10,000 gold coins and the title of Thane in my court. Signed the Nobleman.'

"On second thoughts, I think I'll just claim the 4000 now." You decide.

The problem is, this makes sense in the real world, where consequences are a thing, and you can't just cut people down in the street. But in the context of a game keeping the armor (if it is actually usable and good) and getting more xp (and loot) from the follow-up hit squad is just a blatantly obvious thing to do. There is no downside. And then when you've outleveled the armor and replaced it with something better, you can go back and turn it in for the reward.

I don't really want to see 1000 people running around with the same UNIQUE item, but maybe for some minor items.

actually thats true. Maybe for some sort of amulet or ring of some sort.

Quest items in MMOs are rarely equipable so we are not going to see this I shouldn't think.

If I see ten thousand people with the wabbajack I'm going to scream

As much as I would hate to see that, seeing hundreds of people turn into goats in Cyrodiil would be very amusing.

Hahaha! Always look on the bright side.