> It would be nice if there was a built-in customization which caused
> *Completions* to update as you type, as long as that buffer is visible.
> I imagine such a request has been made before, so what is the obstacle
> to adding it?

I don't remember what was the obstacle, but here is the previous patch
that implements the behavior of zsh and is based on icomplete-mode.

> I would like to figure out a solution which everyone is happy with, and
> then I would be happy to implement it.  It seems to me that it's just a
> matter of, after each keystroke, triggering (for minibuffer completion)
> minibuffer-completion-help or (for buffer completion) some new function
> to populate *Completions* with all-completions output.

There is one difference between icomplete-mode and zcomplete-mode:
the former uses completion-all-sorted-completions where the
recently used items are at the top of the completion list.
Whereas the latter uses the same alphabetical sorting as TAB
shows in the *Completions* buffer.  Maybe a new function
should allow any sorting order?

You are welcome to implement such a new function
to populate *Completions* with all-completions output.
Currently I have no idea how to do this.

> This could (I guess) be done with after-change-functions, although
> maybe others have a better idea.

icomplete-mode uses post-command-hook, so zcomplete-mode does the same.