The Doctor (Peter Capaldi) and his android companion Nardole (Matt Lucas) are living on Earth as a university professor and his assistant. Bill Potts (Pearl Mackie) is called to the Doctor’s office because he notices her monitoring all his lectures. He takes her on as his advanced physics student. Bill develops a relationship with a student named Heather (Stephanie Hyam) who has what seems like a star in her eye. Heather asks Bill to look at a puddle to see if she sees something strange about her reflection. Bill thinks there is indeed something strange but doesn’t know what. When Heather leaves, a voice says that the search for a pilot has begun.
Later, they go back, and Heather vanishes. Bill asks the Doctor to look, and he says two things—the pool is not water, and the image is not a reflection but a mimicry. Back in her flat, Bill is chased by a fluid which takes Heather’s form. She runs to the Doctor’s office, and they are chased into the TARDIS. The Doctor moves it somewhere else, where he and Nardole check on a mysterious vault they are guarding. It is clear that the image of Heather is only interested in Bill. It turns up again, and they move the TARDIS to Sydney, Australia.
The Doctor confesses to Bill that he is an alien. Heather arrives again, having crossed the world in a minute. The Doctor evacuates to another planet a million years in the future, but the image turns up again. He thinks it is part of an alien ship and it chose Heather as its pilot because she wanted to run away. The Doctor makes it pursue them through a battle between the Daleks and the Movellans, and it turns up in the shape of a Dalek. Bill convinces it to let her go and it disappears. Back in his office, the Doctor tries to wipe Bill’s memories, but she protests, and he agrees to take her with him.
The episode received positive reviews, and Bill Potts was liked a great deal. The Doctor lectures on Time and Relative Dimensions in Space. He has lectured there for fifty years. In his office, he has photos of his wife River Song and his granddaughter Susan Foreman. Also, he has a collection of sonic screwdrivers in a cup, like pens. Bill mentions people being possessed by lizards in the brain, which appears to refer to the series Stargate.