This was the first Christmas special with Steven Moffat as executive producer and Matt Smith as the Doctor: it was quite a pop cultural mashup.
The most obvious riff is the permutation on Dickens A Christmas Carol. Moffat added a twist of more far-reaching time travel, as the Doctor tries to change Scrooge/ Kazran by being his Ghost of Christmas Past and giving him a better past so that he will be a better person in the present and help a ship that’s in trouble.
I thought it was delightful, capturing a spirit of “Christmass-ness” in many ways, from the Doctor arriving via chimney, to the giddy joy of "Merry Christmas" that he and Kazran Jr. greet Abigail each Christmas Eve, to the Christmas Eve dinner they share with Abigail's sister’s family.
The mashups came as smaller plot points: the Doctor spending one Christmas Eve at Frank Sinatra’s lodge where he becomes engaged to Marilyn Monroe; the bridge of the spaceship that’s in danger looking a lot like the Enterprise; Abigail (the Welsh singer Katherine Jenkins) singing Holst’s "In the Bleak Midwinter."
But my favorite is that the “monster” of the episode is an enormous shark. When it tries to kill the Doctor and Kazran Jr. it looks like it’s straight out of Jaws---all that’s missing is a quip about needing a bigger boat (Moffat doesn’t seem to know that the line is de rigueur for that trope.)
But more than that, the Doctor tames the shark until it is dolphin-like, and then hooks him to a sleigh for a fanciful flight over faux Victorian London.
Hmm. We know that Moffat chafes at all the Russell Davies particulars that he inherited. In this episode he has the all powerful psychic paper short out-- with the “serious and mature adult credential” being a lie that is just TOO big for it--- and he chips away at the even more powerful sonic screwdriver, as the Doctor asks “what good is the screwdriver” when half is stuck in the shark.
Is it possible that Moffat resents needing to provide a special Christmas episode, which as Ross Ruediger pointed out to me, only started with the reboot? Methinks the flying shark would suggest so. You just don't get to use a shark as a character without the "jumping the" baggage. Moffat, a consummate tv writer, would know that.
Nonetheless, even with the playful jibe, he has added a memorable episode to the Doctor Christmas canon. Casting the great Michael Gambon was an enormous gift itself, as are the songs from Jenkins.