There have been many questions on meta about how to handle bad questions, the people who ask them, and the people who answer them.
We've tried nagging the user before they enter the question. For example if you start to type a question you'll get a list that says "Hey, some of these questions might provide you with an answer."
And the offenders simply ignore it, right?
So lets not let them ignore it. Let's make them look at and respond to the duplicate questions. Yes they will have to work harder to ask their question, but that's the point!
If a poster starts to ask a question that the system thinks is a possible duplicate, display the list of possible dupes and force the poster to respond explicitly to each one (up to a reasonable limit -- say 10). Make them explain why the "duplicate" question does not meet their needs, and make these responses part of the permanent record for this question.
Doing this up front will have a couple of benefits.
First because they are forced to actually open the page containing the possible duplicate question and its answers they may find what they are looking for and not post the question in the first place. (It might even make sense to reward them for agreeing "Yes my question was going to be a duplicate, so I did not ask it." Say the first couple of times they do that they get some bonus rep (not to often or this, too, will be exploited by rep-hounds.)
Second, if their question is truly addressing new ground, they will have already gone through the process of explaining why the previous questions don't address their need. During the process they may clarify their question to the point where it's is obvious why the new question needs to be asked.
Finally, if they try to blow by this and don't provide reasonable justification it should be much easier for others to down-vote, flag, etc. their question. No need for a dialog. The usual "this is a duplicate", "no it's not" dialog will have already happened -- and only the OP will have to invest time in it.
In egregious cases, we could even impose a more severe penalty for questioners who don't respond appropriately to the "is this a duplicate" challenges. Maybe significant loss of rep or quicker question bans at the discretion of moderators.
Bottom line, rather than trying to penalize bad questions after the fact, make them harder to ask in the first place.
[implementation detail: There should be a way to view the other questions and their answers without losing track of the page on which the new question is being submitted. Like any good "idea man" I leave that to the ingenuity of the developers (grin)]