Spinksville has a great post up about the great benefits and joy of duoing in MMOs, or playing alongside one other person as you level your way up. I haven't been able to duo in a while, but I have done one character with a good friend -- he played a Priest and I tanked on my Warrior, and we shot up through to max level (even before the leveling changes) just because not only did our characters compliment each other, but we knew each others' play styles and were able to take on lots of big mobs and quests without a problem. I agree with Spinks: playing the game alongside someone you know well is the perfect mix of a singleplayer and MMO game.
There are certainly issues -- if one of you plays more than the other and pulls ahead or falls behind, it's not quite as fun for either of you. But especially if you're both playing alts and you've got good, regular amounts of time to play together, it can be really fun. In fact, it's kind of surprising that Blizzard and other MMO companies haven't gone in for more of the very-small group play -- singleplayer instances got a pass, but then again, phasing has changed things since then, and the LFG system has improved, so maybe it's time for Blizzard to put some two-player content in the mix.
It would definitely be great to see some official support for two-player content -- not everyone has a partner to play with all the time, but I think you'll find way more two-player groups in the game than you will find full raids of 25 people, and that group certainly gets their share of things to do. Spinks makes some great suggestions about duo content, and it's definitely a realm of gameplay that a lot of MMOs haven't yet officially explored.