After watching the video above, I don't think there's any doubt that the Coen brothers intended the parallels between The Dude and Alice in Wonderland.
- both start out without any ambition or responsibility
- "fuck it" attitude
- do drugs that change their size
- end up in wonderlands (rabbit hole & toilet bowl)
- Bunny, who is white, has a license plate LAPIN which is "rabbit" in French. The Dude is also chasing his own white rabbit.
- Alice finds a golden key which becomes large as she shrinks. When Dude arrives in Lebowski's huge home, Brandt shows him the large gold key to the city of Pasadena, and for no real reason
- Tweedle Dee and Tweedle Dum are Wu and the other guy who show up at The Dude's house in one of the first scenes of the movie
- Walter and Donny are the Walrus and the Carpenter
- Maude is the Red Queen
- Treehorn is the Mad Hatter
- The Nihilists are the Red Queen's henchmen.
What does this all mean?
Well if you consider the parallels, the meaning of the movie changes. If Maude is the Red Queen, then she's masterminded the entire plot. That's what she's laughing about on the phone with Knox Harrington. This is how the idiot, incompetent nihilists are able to pull off a fake kidnapping - Maude is instructing them.
The underworld The Dude enters is a matriarchy where the women control everything and give men (such as The Millionaire Lebowski) the illusion that they have any kind of control.
So what? So nothing, Dude. Just kind of interesting, wouldn't you say?