Doctor Who: The 10 Best Alternative Universe Doctors
Shalka Doctor: Another alternative Ninth Doctor was played by Richard E. Grant in "Scream of the Shalka" was a one-off flash-animated special celebrating the show's 40th anniversary. It was produced by Cosgrove Hall and broadcast on the official BBC Doctor Who website. It's one of the few officially recognized animated outings for The Doctor, and it's also by far the best. In it The Doctor saves a small sleepy town from aliens that haunt it, inspiring terror with their screams.
To get a glimpse of the Shalka Doctor you have only to imagine Grant as The Great Intelligence. He brings much of the same snideness to the role, but uses that same powerful expression tempered with an iron kindness. Interestingly enough, The Master is his companion, having been implanted into a robot body in order to earn atonement. Plans were made to expand the six-part serial into a full-length film, but were scrapped when the reboot series was greenlit.
See also: Doctor Who: A Regeneration FAQ
The Valeyard: The only alternative Doctor to appear in the series proper (Unless I'm right about the Hurt Doctor) is The Valeyard played by Michael Jayston during the Sixth Doctor's time. Unknown to the Time Lords, the Valeyard was a future incarnation of The Doctor seeking to frame his past self for crimes and ultimately execute him in a plot to steal his regenerations.
The Valeyard is one of the biggest headaches when it comes to establishing the canon of Doctor. Though name-dropped in "The Name of the Doctor" there is no telling if he is some sort of concrete destiny for The Doctor, or merely a possible one. Regardless, he represents the darkest parts of The Doctor, and one of the reasons he says, "Good men don't need rules... today is not the day to find out why I have so many."
Exile Doctor: Before Peter Capaldi was announced as the Twelfth Doctor the biggest topic for debate was whether or not we would finally get a non-white or (Gasp!) a female Doctor. The thing is, we've already had a female Doctor. Not only that, she was an American!
Or at least Arabella Weir was born in America. She played another of the Unbound Doctors in audio stories, "Exile." Her Doctor was on the run from the Time Lords, and had recently regenerated into a female after her previous incarnation committed suicide. Her Doctor is one of the most melancholy of all the Doctors, frequently drunk and hiding in exile on Earth to escape pursuit from Gallifrey. She was prone to talking to her past selves while drinking. Nonetheless, when called to act she too stood against evil.
The Other There are hints that The Doctor is more than a Time Lord, more than even the last of the Time Lords. During the Seventh Doctor's run script editor Andrew Cartmel began to lay seeds throughout Seasons 25 and 26 for something called Cartmel's Masterplan that would establish The Doctor as one of the three godlike figures that founded Time Lord society.
The Masterplan never came to fruition on television, but parts of it survived in the New Adventure line of novels. If you only ever pick up one Doctor Who novel make sure it's Lungbarrow. The book offers strikingly different looks on what we currently know of Gallifreyan society, and strongly connects The Doctor to this mysterious and powerful figure from the planet's past. The Other is also mentioned in the novelization of "Remembrance of the Daleks," where text reveals that The Other may not even have been from Gallifrey at the beginning. If he is indeed a Doctor, this could finally resolve the hint the Eighth Doctor dropped that he was half-human on his mother's side and explain his affection for Earth.
On the other hand... maybe it's just one more story amidst the thousands that surround The Doctor wherever and whenever he goes.