Who Am I? on the mysterious past as plot hook.
In order to blatantly trawl for readership over there, how might one apply mysterious pasts to Doctor Who?
For starters, plenty of characters have had secrets, starting with the Doctor. (What his secrets are has changed as the series progressed, but never mind.) We've had public school boys who were really alien agents of infernal powers, girls who never used their real names or talked about their families, women who know the Doctor's future...
But looking specifically at things the player might not know either... the Whoniverse is big and odd enough that an orphan could discover he was an alien, a Time Agent could have a year of his memories deleted by his employers, a bride could appear on the TARDIS in a flash, another orphan could discover she wasn't an orphan at all before time started to crack, and the boy who waited might really be made of plastic... for a while, at least.
If the players give you free rein, their characters could be anyone and anything.
If any of the players take the Amnesia trait or say "grab me with this hook", it might be best to discuss some limits and vetoes, but any crazy scheme could work.
And if none of them do, you could still do it for a session or so, as long as you reset things afterwards. Probably best to ask the player before revealing the PC is really a Flesh duplicate and the real PC is heavily pregnant and kidnapped months ago, though....
The main issue with a mysterious past in a game about time travel is that you can very possibly go there.
Example: The Boy With No Name
Back in his home time and place, Rajiv finds that nobody recognises him. This is the right time, but someone else is living in his family home, and while his neighbours have not changed, they don't seem to know him at all. Has he been taken out of time? How, and why? And if so, how long until this catches up with him?
Mad Example: Presumption
Miss Emily Stevens finds herself invited to a ball at Mortmere House, tomorrow on the Sixth of August 1811. This is somewhat surprising, as she quite distinctly remembers growing up in the early 21st century, not two hundred years earlier.