Is the 11th Doctor the Mad Hatter? Take this excerpt from Wikipedia.
Appearances in the Alice books

The Hatter and March Hare try to put the
dormouse in a teapot
The Hatter explains to Alice that he and the March Hare are always having tea because, when he tried to sing for the Queen of Hearts at her celebration, she sentenced him to death for “murdering the time,” but he escapes decapitation. In retaliation, Time (referred to as a “Him”) halts himself in respect to the Hatter, keeping him and the March Hare stuck at 6:00 forever. As such, he exclaims “Tea Time!” at random occasions. The tea party, when Alice arrives, is characterized by switching places on the table at any given time, making short, personal remarks, asking unanswerable riddles and reciting nonsensical poetry, all of which eventually drive Alice away. He appears again as a witness at the Knave of Hearts’ trial, where the Queen appears to recognize him as the singer she sentenced to death, and the King also cautions him not to be nervous “or I’ll have you executed on the spot.”
When the character makes his appearance as “Hatta” in Through the Looking-Glass, he is in trouble with the law once again. This time, however, he is not necessarily guilty: the White Queen explains that quite often subjects are punished before they commit a crime, rather than after, and sometimes they do not even commit it at all. He is also mentioned as being one of the White King’s messengers, and the March Hare appears as well as “Haigha”, since the King explains that he needs two messengers: “one to come, and one to go.” Sir John Tenniel’s illustration also depicts him as sipping from a teacup as he did in the original novel, adding weight to Carroll’s hint that the two characters are very much the same.