It is all nonsense of course but her statement has enough narrative weight to trigger the opening titles. The viewer is meant to at least speculate about that shocking statement. Could it possibly be true?
It would be an amazing twist if the ‘companion’ had been the Doctor all along. It could also explain Clara’s strange behaviour as discussed here. Missy could have been foreshadowing that the Doctor had also changed gender.
The Doctor disguising his identity is a natural development of the 11th Doctor’s efforts to keep a low profile. Having deleted references to himself from databanks he then makes sure that no one thinks of him as the Doctor by assuming a new female persona.
The big question would be if Clara is the Doctor who is the 12th Doctor? This is particularly tricky as we see the 11th Doctor regenerate into the 12th. It could be that what we see on screen isn’t the literal truth and is only what Clara wanted people to believe. This would seem to be a cheat on the audience but necessary if the 11th Doctor stories also occurred without Clara (because she didn’t come into existence until he regenerated).
If the 12th Doctor isn’t the Doctor she’d need someone to play the role. It could be that he was a native to Trenzalore. The new Doctor would only have to introduce him to old friends of the Doctor as a new regeneration, explaining his confusion, to be believed.
Clara/Doctor could fly the TARDIS remotely or pre-program it to make the fake Doctor believe he was piloting it. She spends some time away from the TARDIS to convince all others around her that she isn’t the Doctor, recalling her time machine when she is ready to travel again.
If the 12th Doctor is a fake it could explain his confused behaviour. His rambling and doubts about his identity could all be because he is just a decoy, brainwashed to think he is a Time Lord (maybe using the TARDIS telepathic circuits to download some old memories).
Alternatively Clara/Doctor could be a later incarnation of the Doctor, doubling back on his/her time stream. The 11th Doctor apparently sensed that this was going to be a particularly bad regeneration.
Without Clara it could be that the 12th Doctor made terrible mistakes. Without someone to act as his conscience what would he have done? Subsequent incarnations might deeply regret the actions he made.
‘The Day Of The Doctor’ illustrated that future Doctor’s can help their past selves. This might give a female Doctor the idea to go back and become his own companion. She could then fix time from behind the scenes. This would tie into the 12th Doctor’s vow that he was going to do something about his past mistakes.
Her paradoxical nature could be why the TARDIS disliked Clara. This would mean that that Clara didn’t need Missy to give her the Doctor’s telephone number. It could be that Missy was lying when she claimed to be responsible for pairing them up. Of course Clara/Doctor doesn’t need to be the original.
There are two events in which a future Doctor could have swapped places with Clara, maybe having regenerated to specifically look like her. First this could occur in ‘The Name of the Doctor’ when the 11th Doctor attempts to rescue her. He might instead find his future incarnation and retrieve her instead (there are plenty of incarnations walking around.)
Secondly it could occur during ‘The Day Of The Doctor’. It could be that it is Clara/Doctor who successfully rescues the earlier Doctors. This is a point when all points in his life merge, including that of the 12th Doctor. One more future Doctor could be possible.
A substitution would explain why Clara has a family. Clara/Doctor simply keeps the ruse up, allowing her to blend in with humanity. Was her relationship with Danny also part of her cover story or did she find she actually had feelings for him?
If the Doctor is only posing as Clara it could explain the frequent occasions in which she doesn’t know things she really should. This future Doctor is attempting to play dumb but forgets what the real Clara should know.
The fate of the original Clara would need to be decided. Her statements in ‘Death in Heaven’ indicates that she doesn’t exist any more. Either she fractured in the time stream or died at some point.
Might the 12th Doctor begin to suspect the truth? Does Clara/Doctor have her own TARDIS if she is from the future? If the 12th Doctor is a fake what happens when he is critically injured and doesn’t regenerate? What if Clara/Doctor regenerated before his very eyes?
Time Lord PCs might try to pretend to be their own companion or create a decoy. This adds an additional complication to a normal adventure. Not only do they still have to save the day they have to do so without tipping their hand.
If a Time Lord does travel back along his own timeline they might attempt to be several of their past companions. This could alter their own history but depending on the original timeline this might be worth the risk.
Earlier incarnations of themselves could visit an old companion only to find out that they only remember a few of their adventures. The investigation into who was impersonating them could lead into their own future, revealing the deception. Do they prevent their future incarnation from enacting their plan and risk changing their history again?