Dr. Elizabeth Klein was an inhabitant of a defunct alternative timeline accidentally created when the Seventh Doctor and Ace left a piece of laser technology in the hands of the Nazis at Colditz Castle in October 1944. This led to their victory in World War II. She was the sole remnant of that timeline and a companion-in-adversity of the Seventh Doctor.
Biography
Klein was a favourite of the Führer, Adolf Hitler, who assigned her many important duties on behalf of the Reich. In 1957, she was commissioned to investigate the outbreak of a deadly virus in Kenya. She failed to trace its source and was pulled off the mission before she could investigate further. This failure would haunt her as the years went by.
In 1965, Klein was working in the physics department at Cambridge University. She was recruited by Jonas Faber to analyse alien technology for the Reich. She became romantically involved with Faber, who gave her access to a device which had been kept locked away from everyone – the TARDIS. She was told its pilot had been fatally shot by a border patrol in 1944, after which his body had been stolen. Klein could not enter the vessel until a mysterious man calling himself "Johann Schmidt" gave her with the key.
Schmidt was an alternative version of the Doctor’s eighth incarnation who, claiming to be a fellow scientist, fabricated a "flight log" for the TARDIS to allow Klein to pilot the ship back to any of its recent destinations. Schmidt manipulated her into travelling back to 1944 to retrieve the Doctor to make him teach her to pilot the TARDIS. Fabe saw through Schmidt’s ruse, but not before Klein had decided to disobey his orders and make the trip into the past. She left her lover behind, promising to make him Führer by her efforts. (BFA: Klein's Story)
Unfortunately for Klein, her arrival in October 1944 alerted the Doctor to the impending change in history and he prevented it. Klein was stranded in the proper timeline. Believing her mistake had destroyed the correct version of history, she vowed to get the TARDIS back and restore her world. (BFA: Colditz)
With other Nazis who fled the fall of Berlin in 1945, Klein ended up in South America. She worked with Hans De Flores and earned an additional doctorate, this one in medicine. She set out for Africa, where she hoped her timeline's events would repeat in this timeline's 1950s. This would give her another chance to learn the source of the mysterious virus. Isolated in a house with other women during the Mau Mau Uprising in British Kenya in 1953, she again met the Seventh Doctor. Despite their animosity, they worked together to solve the mystery of the virus and stop an alien plot to use Earth as a testing ground for biological warfare. The Doctor, knowing Klein would continue to pose a threat to history, insisted she accompany him as a companion on his travels so he could keep an eye on her. (BFA: A Thousand Tiny Wings)
After Klein and the Doctor left Kenya, he asked her for the story of how she came to pilot the TARDIS to 1944. Upon hearing the details, he explained to her that the enigmatic "Johann Schmidt" had been an alternative future regeneration of himself. Klein was furious to learn her world had been lost through her manipulation by the Doctor, but pretended to accept his version so she would have a chance to turn the tables. (BFA: Klein's Story)
She travelled with him for a lengthy time and learned how to control the TARDIS by watching him. Her chance to betray the Doctor came when he lost his TARDIS key while trying to help a Vrill warrior hatch from its egg. Klein recovered the key and abandoned the Doctor on the Vrill planet, fleeing in the TARDIS to restore her own timeline. (BFA: Survival of the Fittest)
Klein tried to use the TARDIS to restore her exact timeline, but failed. She created what she described as a far better one. The details of the changes she made to history are unknown, except that the new Reich began in 1944. She avoided trying to bring back her lost love, Jonas Faber, because she knew she would never be able to perfectly restore the past they had lost. By 2044, Klein had used time travel to defeat of many formidable alien threats, including the Sontarans and the Daleks.
Though the Doctor’s past had been completely rewritten, his memories of his experiences with Klein resurfaced in 2044, replacing those of his alternative self. Before this, he had arranged for an attack by Selachians from the future on Earth's moonbase to lure Klein and the TARDIS there. The Selachians used future technology to destroy the TARDIS, separating its internal and external dimensions and stranding Klein in the invasion before she could prevent it by equipping the past with knowledge from the future.
The Doctor told Klein she had been tried (presumably by the Time Lords, though he did not mention them by name) and took her out of the timeline with another TARDIS specially equipped to carry out her sentence. She willingly submitted to her punishment – total obliteration from time – hoping it might restore her original timeline by preventing her from ever meeting the Doctor.
The actual outcome was quite different. The prime reality was restored and another Dr. Elizabeth Klein came to be. This was a Klein known for her triumphs working for UNIT. Her personality appeared to be the same, except that this new Klein devoted her talents to the protection of Earth. Upon meeting the Doctor in 1962, she recognised him slightly although her experiences with him had been erased. (BFA: The Architects of History)
Personality
Klein was a fervent believer in National Socialism and the benefits of fascism in general. Although she held that the needs of the many outweighed the needs of the few, she formed close, compassionate relationships people, as evidenced by the impact Lucy's death had on her (BFA: A Thousand Tiny Wings) and her concern for the Vrill (BFA: Survival of the Fittest). She knew of the atrocities which Nazis like Mengele committed, but differentiated their actions and ideals, saying that Mengele went too far.
Even so, Klein was not beyond coolly executing an enemy or torturing one for information. She was not intimidated by the Doctor nor his reputation. She was not a hypocrite; she was willing to die for the greater number of people to survive, although she admitted she would prefer to live.
Rather like Liz Shaw, Klein was recruited from university. She showed an interest in and knowledge of the more esoteric and experimental sciences, such as time travel. Faber said she was unlike many of her contemporaries, with an open, questioning, mind. Before her removal from history, she came to agree with the Doctor on some points, particularly with his observation that no matter how much she changed history, it would never feel completely "right".