Can an Markov decision process be dependent on the past? As far as I know MDP are independent from the past. But the definition says that the same policy should always take the same action depending on the state.

What if I define my state as the current "main" state + previous decisions?

For Example in Poker the "main" state would be my cards and the pot + all previous information about the game.

Would this still be a MDP or not?
It's not totally clear from your description, but it sounds like you may be onto something like an Additive Markov Chain.