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Batman (1989):
In the final confrontation between Batman and The Joker, the conflict ends with Joker falling into a cauldron of liquid that would surely kill him, only to be saved by Batman, who holds him by a limb, thus stopping his fall.
This scene is virtually identical to a scene in the 1989 Batman film in which Batman grabs Jack Napier's hand as he falls into a vat of acid, only to lose his grip, watch Napier fall into the acid and become the Joker.
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The Nutcracker:
As Batman and Robin face off against toys that attack them (including giant wooden soldiers akin to the ones in The Nutcracker), several sections of the Nutcracker Suite are played in a modified chord to reflect the tension of the fight.
The Nutcracker is a famous story that features toy soldiers and is a Christmas staple in many forms of media.
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Betty Boop
The doll Joker plays with has a similarity to the classic cartoon girl, Betty Boop.
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When Joker escaped from Batman's grip he ran and said "Run, run as fast as you can, you can't catch me, I'm the Joker man!"
This is a reference to The Gingerbread Man fairy tale, in which the gingerbread man said "Run run as fast as you can, you can't catch me, I'm The Gingerbread Man!"
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Numerous references to It's a Wonderful Life, the ever present Christmas movie. At the end of the episode, it even shows the two church bells and "The End," which also appear at the end of this movie. At one point during the episode, Robin mentions that "[the movie] is about how much difference one person can make" and asks Batman if that sounds familiar.
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Robin: He could give lessons to Scrooge.
Ebenezer Scrooge from A Christmas Carol is a fictional story about an old and bitter man, who undergoes a profound experience of redemption when he is haunted by ghosts on Christmas Eve.