This past season of 'Doctor Who' we saw the Doctor wail away on his electric guitar in several of the episodes. From outside the reality of Toobworld, we know this wouldn't have happened if Peter Capaldi wasn't a proficient musician skilled in the instrument. At one time he even had his own band... and Craig Ferguson was also a band-mate.
But was it Capaldi's incarnation of the Doctor who learned how to play the axe? There was plenty of "time" between several episodes; they didn't just flow from one into the other.
However, I think it was an earlier incarnation who took the time in the TARDIS to pick up the skills, perhaps even calling on a few celebrated musicians for lessons. Jimi Hendryx, for example?
I think the best candidate would be the "Ninth" incarnation (actually the Tenth due to the War Doctor). I think he best had the personality to turn to music to help heal his psyche after the pain caused by the Time War.
And in one of those off-screen adventures, maybe the Ninth Doctor formed a band. And who would be in that band?
Marcus Aurelius, Roman emperor, last of the five good'uns, stoic philosopher.
Superlative bass guitarist. The Doctor really knows how to put a band together.
And this would be the perfect story to bring super "Who" fan Craig Ferguson as the Doctor gets the band back together for an adventure. Perhaps something involving an alien threat in 1970 that ended up killing Jimi Hendrix, Jim Morrison, and Janis Joplin (who previously gave the Doctor the great-coat which he wore in his tenth incarnation.)
As for Craig Ferguson, he should be an alien member of the Doctor's band. And for some reason, I'd like to think of him in the make-up of a Tivolian, as seen in "The God Complex" and "Under The Lake" & "Before The Flood". Perhaps it would explain the Doctor's dismissive attitude towards the funeral director - a Tivolian would probably cave in rather than try to win a battle of the bands.
Just a suggestion, Fanficcers.