I just posted a question that I may want to expand on if answered, depending on the answer. Should the new questions I have be posed within the current question, or should a new question be posted?
First off, I'd like to say thank you for taking the time to learn how this site works. It's always great when a new user takes the time and effort to interact constructively with the community here. So, kudos to you for coming to Meta with your question so quickly!
Now to actually answer your question. One of the rules of thumb we have is that we prefer that questions not be edited in ways that invalidate existing answers. Very few people like trying to hit a moving target (i.e. answering a changing question). So, if you have clarifications to make, go ahead and make them, and earlier is better.
If the answers inspire another question, like if you want to know more of the details, then I would say ask a new question. If your original, stated question has been answered, that's great! Leave it be, and ask a new question to continue your line of inquiry.
I agree with El'endia on this point in general, but I'd like to make sure one thing is clear: you have a right to expect answers to your questions to attempt to fully answer your questions. Don't ask a new question if its answer space should be well within the answer space of the original question; push answerers to be more thorough if that's the case.
Otherwise, if you have a genuinely different question, even if it's closely related, post away! Sometimes it's a bit of a grey area whether a "closely related" question is properly its own question or a subset of one that's already been asked. If you find yourself in that grey area, reread El'endia's post, take a few deep breaths, and follow your conscience. :P