I meant like on Wowhead, you can pin your main character and when you look after quests it weights the results based on that character, hides irrelevant data, shows you relevant information. Not saying Wowhead is the best either, but what I meant was "putting your character in the center and showing information relevant to that character" if you know what I mean.
In-game you can "show only items usable" but this could be extended even further with a website, it could consider current gear, content (based on achievements/gear), e.g. show the best possible item upgrades based on what you got and where you are at currently, hiding if you like items you already own*, quests done rewarding items you can't get again (if the gear improvements suggest item rewarded by quest you already did), known mounts and pets could be filtered out on the lists, e.g.
You could have filter options when browsing the AH so you can avoid seeing what you don't need if you are after let's say gear, pets, mounts, profession related reagents, achievement related items required by professions, e.g. You'd be able to make a shopping list based on the cheapest items you can find on the AH and such. It would be a very powerful system I reckon.
The problem in-game is the query delay, the website can sync the AH (at a interval around 3 hours I believe) and nearly instantly update all available items and also it can sync character info, also with the new API.
Many many many possible features that can be implemented! ^^
Edit; oh, also sites like askmrrobot can make it so gear improvements that you need also provide a "AH buy list" of some sort. For example you need new gems for 6 items, it can check the AH for what items are available and tell you what to go and buy. I am sure there are other scenarios where the AH can be used to help tell the player what to do next, what to buy next.
When talking about achievement and professions earlier, for example you wish to get The Northrend Gourmet and you gotta make 45 recipes, it can check what you already got, what you need, your professions, tell you if the AH got what reagents, how many to buy, e.g. I keep getting ideas for features such a system utilizing the API can provide players with. They keep coming, make them stop!