I'm suddenly more interested in starting my search back up and I'd like to include a few more options this time around. Unlike my other suggestions, these are things that would need to be hashed out prior to starting an RP since they'll have to manifest fairly quickly.
-The Doctor, being a professor at a university, is going to have a slightly more conservative attitude, just like he did when he first used the device to become human (in Human Nature/The Family of Blood). As such, I would suggest your character be the one to come onto him rather than waiting for or expecting him to with her; even after awakening as the Doctor he's going to be slightly more apprehensive about taking a companion with him due to what led him to being there. If she has a more forceful attitude (just a touch moreso than Amy Pond when she forced a kiss on him), it would help things, so for her to establish that kind of strong personality beforehand would be best. This is not required but strongly suggested.
-It has been suggested that the companion be from a troubled or broken family. This then opens up the possibility for her parents to have surprising secrets, perhaps even being not entirely human. Not by any means required nor strongly suggested, merely food for thought.
-This is by no means a sexual-emphasis RP if anyone was unclear about this, but this Doctor should at least have some romantic interest in his companion, unlike previous incarnations. Yes, this will include physical relations as well as the L-word, but if you wish to omit the old in-and-out, we can fade to black. If you are comfortable with playing out a rough shag with the good Doctor, however, that is also fine by me. If you wish to continue the affair after he has rediscovered his identity, we will then have to explore his personal feelings for her since he'll be partially 'getting to know her all over again' at that point. This does not mean she should ever act like a horny schoolgirl (unless of course she's attempting to get him into bed for the first time; afterwards, of course, that sort of thing will be unnecessary!) We can talk about it if we need to figure out a good balance.
-Obviously this is a British series and everyone of any significance speaks with an accent, and I intend to do the best I can to use British style speech as often and naturally as possible. I would like for you to reciprocate if possible (I find that challenging writing is more fun) but I completely understand if that is not your cup of tea, love. Please just let me know ahead of time if you are uncomfortable trying, and I won't mention it again. Likewise if you have a problem with me trying to imitate a foreign dialect, I shall do my best to make the Doctor sound natural without being too American.
-Please let me know which types of scenarios you prefer; I am perfectly comfortable with virtually anything, from present-day earth (i.e. The Lazarus Experiment) to ancient historical events (i.e. the Fires of Pompeii) to futuristic environments (i.e. New Earth) and even alien planets (i.e. Midnight; that's actually my all-time favorite episode). The one thing I'm kind of iffy about is episodes where they're stuck on a space station in peril (i.e. The Impossible Planet/The Satan Pit, 42). Sometimes they're okay, but usually not. I don't like feeling claustrophobic so that's sort of a turn-off for me, just to let you know. As such, I haven't got any scenarios in that environment in mind, but if you do, I don't mind hearing you out but I will probably want to change some things if I feel 'trapped'. Just a heads up.
-If you are a dude in real life (typically I feel uncomfortable playing sexual RPs with male partners even if their characters are female) then this is your lucky day: Since this is not the normal sexual-emphasis RP I usually go for, I don't mind if you are interested in playing a female character, so step right up if you're interested. I just hope you can play a good female outside of the bedroom, since I would think that is much more complex.