So after I finish the game I will probably want to restart but the only way I would do that is if your character synchronizes to the new game, does this happen or do you have to start fresh?

Starting fresh. This is not a gametype that really has an 'import' type system.

Borderlands did it, was hoping something like that.

Have you finished the game yet? I think you can still explore after the 'game' is over, don't quote me on that though. It would be cool if you were able to do something like 'new game +' or something. Where all the enemies are scaled to your level.

Than again I usually don't play much if i'm offered a new game +. I love starting a new story/new game.

Maybe something the Fallout Team could think about?

There is no New game plus, so you'll have to start a New character if you want a fresh playthrough. You can still explore Boston after you are done with the Main Quest though.