That won't be a good idea for at least two reasons I can think of.
(lack of) Transparency
Stack Exchange is 100% transparent about the activity of its users, meaning each and every action (which is not private, like flagging) is publicly displayed in the user profile: Question, Answers, Edits, and more.
This allows users to track the activity of other users easily, and be careful with what they post, knowing anyone can see anything they do.
Letting users make anonymous actions is against this transparency. Of course one can simply create another account from scratch and use it, but this is already covered in answers to a related discussion.
(avoiding) Tracking of bad behavior
Suppose you see a very poor question from a user. You want to help so you edit, comment, etc. But before doing this, you take a look on that user's past questions. If that's their first or second question you think "New user, let's give a hand" and continue happily. However, if that's their 10th poor question, all 10 posted very poorly, edited, people tried to help, and all in vain, you think "This one is beyond help, no point wasting my time" and move on to help someone with potential to learn.
Letting users post anonymously will prevent you from doing this, making it impossible to know the history of the one who posted, and not knowing if you waste your time trying to improve a poor question or not.