Larísa made an interesting post earlier today about why she thinks that account-wide achievements (a feature that Blizzard has at least hinted at as something that they might look into in the future) are "a terrible idea". I was originally going to reply in a comment, but then I thought that it would just go on and on and that I might as well make a whole post of my own about it.
Basically, I disagree with Larísa on this.
She argues that for example seeing a level one Kingslayer running around would be very immersion-breaking, and that the whole notion of wanting account-wide achievements doesn't seem to be based on anything but a desire to show off to as many people as possible.
I have to say that I found the immersion argument more surprising than anything else, because for me achievements themselves are one of the most immersion-breaking things in the game as it is. Nothing screams "this is a game" like a big golden badge with some witty phrase popping up in the middle of your screen while you're going about your business in the world. The way achievements are recorded afterwards has no connection to the in-game world either, it's just a tab of information accessible on your action bar. Ditto for titles really... seeing "the Kingslayer" next to someone's name can only be considered immersion-breaking if you're also roleplaying that everyone has their name floating above their head, otherwise you wouldn't know anyway.
For me achievements have always been directed at the player, not at the character, because they don't even exist in-character. Sure, you could have your toon talk about that time she faced the Lich King, but that's not really dependent on actually having any record of it, is it? You might as well talk about some completely made-up adventure. Unless you treat your achievements as some kind of magical diary that you carry around in your back pocket and that records things like that one time you fell off the Scryer's Tier while completely smashed... but even so it would require you to make things up that aren't actually supported by the game. Achievements are simply not part of the world, they are something to look at for the players behind it.
With that said, I think it would only be logical if they were actually tied to the account instead of the individual character. I'm not saying we have to severe the connection to the characters altogether, since as Larísa correctly pointed out, doing a fight as a healer can be very different from doing it as melee dps for example. You could have account-wide achievements and still have a note saying something like "completed on: character X", or simply have two layers of achievements, individual character and accout-wide.
Larísa refers to an account-wide achievement system as a "fraud", yet at the same time she cheers for one of her commenters when said commenter says that she likes the way the current system allows her to impress people who think her alt is a noob because it doesn't show up as having any great achievements. Isn't that fraudulent, giving the impression that you know nothing so people won't expect as much from you? I think if anything, an account-wide achievement system would be a bit more honest, letting us know at a glance what kind of player we're dealing with: experienced, new, raider, hardcore PvPer or whatever.
As for the argument that people just want account-wide achievements to show off... erm, of course they do, what else is the point of having achievements at all? Larísa refers to them as "public record or diary", but I think public is the keyword here. They are meant to be shown off. I don't need achievements to remember my greatest moments in the game myself, they are purely there so others can easily see them as well. So what exactly is wrong with wanting them to be as easily accessible and widely visible as possible?
The only downside I see in account-wide achievements is that they would remove one of the incentives to go through old content again. Part of the fun of levelling alts is that rush of fast progression, steady improvement and feeling that you're achieving something (regardless of whether you get a golden badge for it or not). Similarly having to unlock each achievement separately for all of your alts can give you a reason to do a certain bit of content again even if you wouldn't do so on your main. But then, as I said above, having account-wide achievements wouldn't necessarily mean having to abolish character achievements altogether, so there is room for compromise.