Now that the relationship with the new companion is established, it is time to get to what has been going on in the background all season. Nardole, who we first met as a companion to River Song, turns out to be looking after (i.e., monitoring) the Doctor at her behest. It is, not surprisingly, Missy who has been confined to the vault. The oath that the Doctor took is to look after her for a thousand years, the result of his unwillingness to kill her. He is also still blind, so the stakes are high at every level.
Nardole has his own agenda. Missy cannot be trusted. Bill finds the Doctor a bit of an inconvenience, but willingly teams with him when something interesting is happening. And yet, that is just what things appear to be.
The deeper mystery “Extremis” is with the new menace who have created a computer simulation to better allow them to conquer the Earth. The simulacra Doctor manages to figure out the deception and warn the real Doctor.