My view of the Mad Hatter, or 'madness' in general, is not the 'zany' wacky portrayal that Ed Wynn gave and is so often used for the Mad Hatter. It is of a man who's mind has been beaten, broken, and is grasping at any logic he can to give at least the appearance of sanity.
The Hatter (who is only ever called 'Mad' by the Cheshire Cat, who then calls everyone mad) shows none of the signs of "Mad Hatters Disease" (Mercury poisoning). Instead it is stated that he is stuck in an infinite tea party; never being able to make it to dinner. I love the idea that this is a cause (or possibly a delusional product of) the Hatter's mind snapping. Very much in the style of H. P. Lovecraft or Edgar Allan Poe, I see the Hatter clinging to logic, pointless questions, and semantics in a desperate way to keep his humanity. In fact Poe is slightly mentioned with the "Why is a raven like a writing desk" question.
I am trying to make video game worlds filled with that madness; the desperation to keep going further. In my smaller games the players will confront the world trying to crush them, clawing and grasping to survive. In the larger worlds players are faced with the unknown and challenges they have to unpack to solve.