There’s a lot of things I adore about their relationship.
Firstly, there’s the companionship. John has just come back from the war - a traumatic experience. He’s feeling lonely, and he’s poor, and he needs someone to help. Sherlock, we are lead to believe, has had a pretty shitty life himself, and he needs someone to help him be normal, if Sherlock Holmes can ever be such. They need each other - they’re each other’s rock.
Then there’s the more detailed friendship. John always grounds Sherlock. For example, in Baskerville, when Sherlock was ranting about how asdfghjkl; the case was, John simply said “timing!” and Sherlock shut up. John has a hold over Sherlock, and that’s brilliant in terms of keeping a lid on Sherlock’s eccentricities. Of course, Sherlock helps too. He gets rid of John’s cane, for starters, and pulls him away from getting too attached to other people who might hurt him (e.g. girlfriends). That might be seen as possessive or whatever, but I feel it’s more protective, keeping John away from the bad people out there.
Then there’s the bromance. They finish each other’s sentences. They go out together. They read each other’s minds. You can’t deny that they are connected, whether you think it’s platonic or not. Neither of them might be gay, but the fact is that they need each other - they are like two halves of a whole (to quote Martin Freeman).
Then of course there’s the heavily-implied actual romance. “Two bedrooms”. “Your date”. “I knew you’d turn up”. And that’s just in the first episode (Study).
Overall, I think Sherlock and John are a very complicated couple. I see them as everything, from just friends to fucking, depending on the fic or the situation. To answer your question, my favourite thing about their relationship is the fluidity - the way they can go from being BEST BUDS 5EVA to FUCK ME AGAINST THE WALL in about ten seconds. I just really love that aspect.