"Clara" was an instrumental song composed by the Twelfth Doctor. He performed the song several times for Clara Oswald in her TARDIS, disguised as an American diner, during a visit to Nevada and, when asked by her if the composition had a title, identified it as "Clara". This occurred after he erased most memories of his former companion, so he was unaware that he was playing it for Clara herself.
The Doctor didn't know the origin of the tune. However, after discussing the fact that the Doctor no longer had specific memories of Clara, and in particular had forgotten a private message she gave him while in the cloisters on Gallifrey, Clara suggested that songs, such as this one, might represent those lost memories. The Doctor considered this and continued to play the song as Clara's TARDIS dematerialised around him. (TV: Hell Bent)
Behind the scenes
Originally titled "Clara?" and also known as "The Impossible Girl" (the title it was performed under during the Doctor Who at the Proms (2013) concert and the Doctor Who Symphonic Spectacular), the composition was first heard briefly in TV: Asylum of the Daleks in 2012 during the sequence in which Oswin utters the phrase "Run you clever boy, and remember" for the first time.
A full arrangement of the song was first heard on the soundtrack to The Snowmen in 2012, most notably during the sequence in which Clara Oswin Oswald ascends to where the Doctor's TARDIS is hidden; the soundtrack album for the special titles the track "Clara in the TARDIS". The "Snowmen" version incorporates a wordless choral vocal singing the melody, which is unique to this version of the leitmotif.
Prior to this, Gold had also composed "Oswin Oswald", a musical theme for the earlier character. This tune had a similar melody, but a number of noticable differences to the later track.
"Clara?", although introduced for Oswin and Clara Oswin Oswald, became the recurring leitmotif for Clara Oswald proper, beginning with The Bells of Saint John and was featured prominently in the opening minutes of the 50th anniversary special, The Day of the Doctor. It was also borrowed for the Red Button spoof The Five(ish) Doctors Reboot for a sequence in which Jenna Coleman, Matt Smith and Steven Moffat appear.
It continued to be a recurring theme for Clara during Series 8 and 9, before becoming part of the narrative in Hell Bent; a sombre variant was featured during her death scene in Face the Raven. Besides being played on camera during Hell Bent in an electric guitar version by Peter Capaldi, the tune also was prominently featured in the Cloisters sequence where the camera pans away from the Doctor and Clara during their private moment. A narrative link is made between the use of the melody in that sequence and Clara later suggesting the song might represent what she said in the cloisters. Behind-the-scenes footage of Clara's farewell scene, included in the Doctor Who Extra instalment for Hell Bent reveals that Capaldi played the melody off-camera on an acoustic guitar as Coleman filmed her final goodbye to the Doctor.
The theme also plays during a scene in The Pilot, when the Doctor is about to wipe the memories from Bill Potts' adventure with him and she asks him to imagine what it would feel like if someone did it to him. According to Doctor Who Magazine #512, a deleted scene had the Doctor playing the melody on his guitar, on stage in a nightclub as the characters of Bill and Heather meet; the episode, as finally broadcast, removed most of this scene in favour of a montage, with the "Clara" melody removed from the soundtrack. This scene was later broadcast on 1 July 2017 by BBC America, along with a second cut scene in which it is revealed the Doctor now refers to the song by the title "I Forget"; this wiki's policy is to not consider deleted scenes as valid canon sources, so "Clara" remains the only canonical title given to this composition. Had it been kept in the final episode, it would have been a unique case of a companion being introduced as we heard another companion's leitmotif.
This is not the first piece of music attributed (in the fictional universe) to the Doctor; he previously conducted a fourth wall-breaking orchestral performance of another piece of music he created (TV: Music of the Spheres) and throughout the televised and expanded franchise, many references have been made to him either composing or contributing to the writing of other works, including some well-known compositions.
It is also, after "Love Don't Roam" (TV: The Runaway Bride), the second diegetic song to refer (albeit in this case instrumentally) to a lost companion, although "Love Don't Roam", in-universe, just happened to be a song the Doctor heard being played in a night club with lyrics that invoked memories of Rose Tyler.