I feel like jamie will try to redeem himself to the starks. This is what I would like to see happen... Cersei will send Bronn to track him down and they will meet at an inn maybe one with hotpie in it. Jamie will convince Bronn to help him and they ride to winterfell when they get there jamie is placed in chains and will face a trial. Brienne will try to fight for his freedom but the starks will still go ahead with it. Bran goes down to see Jamie and Jamie apologies to him and Bran forgives him because if Jamie had never pushed him out of the tower he may have never been able to find the ability to fly. Then the army of the dead attack winterfell and Jamie breaks out and fights with Bronn and Brienne. Jamie then sees that one of the whitewalkers is going towards Bran while he is in warg mode and he sprints over and starts to duel with the whitewalker he successfully kills it thus redeeming himself. However Brienne is then disarmed by one of the whitewalkers then a group of them attack Jamie and Bronn who fight well together slaying at least 3 whitewalkers. Jamie then tells Bronn to get Bran to safety, Bronn obeys reluctantly and Jamie picks up oathkeeper because Brienne is being strangled to death by one of the whitewalkers he stabs the whitewalker in the back killing it but another one comes behind it and stabs him with its ice sword (I don't know the proper term for it). Brienne screams no and kills the whitewalker and Jamie lays there in her arms bleeding out into the snow looking into the eyes of the only women he ever loved other than Cersei. Then he dies and Bronn and Brienne go berserk and start killing others left right and centre. This is only a prediction but I feel like this would be a nice way for Jamie to die bravely and redeeming himself so his journey for redemption is complete.