Man, I just came up with a nice word-saving, time-machine fix for Possession.
My time-machine fix is a bigger Smugglers. Like how Fortune is a much simpler Outpost. Instead of processing another turn with their deck, let's look at the results for a turn that just happened.
Possession: Action, $6 [P]
For each card the player to your right gained in their last turn, gain a card costing as much or less.
Hmm, well as Holger pointed out, that doesn't fix Possession's stalemate problem at all. I like it in general, but it still needs "If this is the first time you played a Possession this turn" clause. Historically, your advice for countering Possession has been, "don't build a deck that can buy multiple Provinces in a turn", but that implicitly assumes your opponent only plays one Possession per turn. In reality, you want to play multiple Possessions in a turn. And usually the only reasonable counter to a multi-Possession deck is to make your deck awful; Victory cards help you, but not your opponent.
My fix preserves the joy of taking somebody else's turn, but some people probably hate that extra-turn aspect. I guess the question is: how much Possession hate comes from different aspects of Possession. There's
1. I hate that the extra turns make the game drag.
2. I hate that somebody else gets to benefit from my deck.
3a. I hate that the only sensible counter is to make my deck awful.
3b. I hate the stalemates that happen when we're both playing multi-Possession decks, so we both make our decks awful.
4. I hate that some cards become dangerous traps in a Possession game (TfB, Cartographer, etc.).
#2 is baked into the concept of the card, and so much for that. Your fix removes #1 and #4, but leaves #3 intact. And it adds a significant tracking component. My version fixes #3 and some of #4 (TfB is now a counter rather than a trap), but leaves some of #1 intact (but not all since multiple Possessions don't stack).
Anyway it's a tough call. Personally, I think the joy of taking a turn using another player's deck is worth preserving, but your mileage may vary.
I do think you want it to trigger on gains from last turn, not next, otherwise the next player has a pretty unfun choice to make.
I don't think making Possession 2.0 act on the previous turn's gains changes the incentives for the players, it just moves the dilemma up one turn and makes it a little speculative.
I think the duration effect idea is pretty elegant. I think it's good that it has some element of player interaction.
I think you're implicitly assuming expert players here. I will bet that casual players would not like having Duration Possession hanging over their head, making them not want to buy good stuff. You can already see some of that with Swamp Hag. I do like that it removes the tracking component of Donald's version, but I don't think that makes up for the bad feelings.
EDIT: In other words, I agree with pacovf.