Edit: I've lately had doubts about how much Columbia and Brad's relationship (or those of others I've mentioned) shows the possibility of helpful, understanding, and connecting relationship. Different things in my personal life have led me to this. They seem mesmerized be other, and there's an underlying sense to me of a sort of mother-son relationship that has a certain sexually consuming desire to it, a certain coveting of the others surface role as an unconventional risk taker versus innocent, repressed boy.
What I originally wrote:
I'd be really sad if there wasn't a romantic connection between them, since I relate most to Brad, and I find Columbia the most attractive of the women in the film (especially when her makeup starts to wear off in the end).
I think a case can be made, however, but their connection is only very lightly hinted at. If you've ever watched
The Girl with the Dragon Tattoo (the Swedish version), you'll notice that there is very similar dynamic in the first meeting of Blomkvist and Salander and Columbia and Brad - his attempts at standard greetings, her seeming to laugh at such conventions. I've yet to fully understand Salander's reasons for loving Blomkvist, but it is explained in the book she has protective feelings for him because of how naive and easily trusting he is - how much of a "dope" he is. Columbia acting like a mothering figure for Brad also reflects on this. If you find it hard to swallow the idea that Columbia is similar to Lisbeth Salander, consider the fact that Joan Jett played Columbia in a 2001 Broadway production of
The Rocky Horror Picture Show.
Brad has a bit of Eddie in him, but I think he is conflicted about that side of himself, just as he is about his sexuality. His more vicious, alienated side seems to be hinted at in his sarcastic and biting comments about the Transylvanians. In Ovid's
Metamorphoses there is a story of the aggressive, masculine Salmacis who pursues the boyish Hermaphroditus. Hermaphroditus blushes and feels ashamed by her declarations of love. The Italian artist Scarsellino hinted at a hidden vicious side in Hermaphroditus by having him pet a black dog with a collar on it:

Another book that has a lot to say about that vicious side and the sort of couple we're talking about is
Jane EyreI think after going through the Medusa Transducer, Brad accessed a bisexual/feminine exhibitionist side, rather than simply a homosexual side.
A few other points of support:
- About Eddy: "I very nearly loved him" (
Nearly.)
- To Frank: "I loved you" (past tense)
- She is in front of the statue of Brad when she says "love for a certain dope"
- When Magenta and Columbia are undressing Brad and Janet, Columbia looks at Brad and says "Slowly, slowly! It's too nice a job to rush."