In contrast to the full TARDIS that seems to be coming up in series 8, I would love to see a bit more empty TARDIS scenarios. It’s not a common thing for the Doctor to not have companions: it only happened once during the classic series (in “The Deadly Assassin”), and once during the modern series (in “Midnight”). I’m not counting any episodes in which the Doctor didn’t have a regular companion but picked up another character that he bonded with closely enough to count as a one-off companion (such as most of the Christmas specials and “Planet of the Dead,” “The Waters of Mars,” The End of Time, “The Lodger,” and “Closing Time.”).
If the Doctor doesn’t have a companion, the episode must focus directly on him or on the situation at hand. In “The Deadly Assassin,” this allowed us to view Time Lord society from the point of view of the Doctor, rather than any alien (to Gallifrey) companion (Sarah Jane, in this case). The script was more streamlined than usual, because the Doctor did not have to explain everything to Sarah Jane, and in the process, the audience got to experience everything, rather than being told what they were seeing. It also allowed the Doctor to get trapped for nearly two episodes in the Matrix without requiring them to keep cutting back to reality to show what Sarah Jane was doing or getting her trapped somewhere so that they could ignore her. The Master also could concentrate on the Doctor, instead of getting distracted by having to deal with her.
The lack of a companion was even more effective in “Midnight.” A story about what happens when humans are afraid of the unknown, “Midnight” would have played out very differently if Donna had been there to try to calm them down and convince them that the Doctor wasn’t the threat. Even if the humans didn’t listen to her (which they probably wouldn’t), the tension of the tight story would have been broken by Donna’s pleas; part of the strength of the final moments of Sky’s possession came from the various characters starting to doubt that the Doctor was the threat and trying to decide if they should intercede.
Both of these episodes came into being under unusual circumstances. For “The Deadly Assassin,” when Elisabeth Sladen left the show, Tom Baker asked for an episode in which the Doctor didn’t have a companion. “Midnight” was series 4’s “companion-light” episode. Back in series 2, in order to expand the series to fourteen episodes instead of thirteen, they created a “Doctor-light” episode, “Love and Monsters,” in which the Doctor and companion appeared only sparingly so that they could be filming another episode at the same time. In series 3, the “Doctor-light” episode was “Blink.” For series 4, they expanded this idea by filming a “companion-light” episode, “Midnight,” with Mr. Tennant appearing in almost every scene, while Ms. Tate was simultaneously filming a “Doctor-light” episode, “Turn Left.” Take a look at this list of episodes: they were all fantastic, with the exception of “Love and Monsters,” which was a fantastic episode until the Abzorbaloff appeared. (Think about how good that episode could have been if a reasonable monster had been the antagonist.) Doctor Who is a great show, but it excels when it steps outside of its usual boundaries.
In my opinion, empty TARDIS or companion-light episodes should be explored more often, to tighten the storytelling a bit, occasionally give the Doctor more spotlight, and take the show in different directions. It’s not difficult to set up – the companion has to go home for some reason, for example – which makes me wonder if it’s a contract thing, saying that the companion must appear in X episodes per series. It isn’t something that should happen often, though – probably not even once a series – but certainly more than twice in fifty years. Perhaps there aren’t many data points, but it seems to be a successful formula for the show, given that the actors who have played the Doctor have all been dynamic performers who could easily carry an episode on their own.