It's important to understand the distinction between a question that takes on new meaning/direction as details are revealed and a related but separate question
In the case of the former, sometimes you start with a moderately vague question or a question that has a lot of depth to it, and then through comments or attempted answers that don't work details are revealed that then subtly influence the question into a new direction that makes the resulting question fairly different from the originally asked question. The "problem" here is that answers that do not track these changes can become outdated and some of these may have been voted up at the time they were placed because they were good answers to the question in the state it was in at the time of the answer.
However, we do not want to encourage folks to simply re-ask the same question with more details even in these cases. Stack Overflow is collaboratively edited for good reason and we shouldn't avoid this just because it can lead to a disjoint relationship between question and answer in some cases. In the case of the former, you should not suggest asking a new question.
The latter is more clear. It tends to take on a few flavors:
- the asker's problem is solved and they have a new question that arises as a "next step"
- the asker's question wasn't really the question they wanted to ask in the first place so even though they may have gotten valid/useful answers, they really wanted to ask something else
- the asker simply asks a completely separate question within the original question
In all of these cases you should suggest a user start a new question. The original question body and tags don't really bear much on the actual issue at hand and are actually detractors from a search standpoint because of all of the unrelated stuff that is being associated with this new question.