I'm running zsh inside multi-term.el (which, as I understand it, is almost identical to term-mode with some additional commands) in character mode.
Edit: emacs 24.5.1 in a terminal (-nw), zsh 5.0.2, https://github.com/ayrtonmassey/config for my full config
When I hit TAB to invoke zsh's auto-complete, if the prompt is at the bottom of the screen the auto-complete options are hidden. This only happens on specific auto-completions: for example, pressing TAB after cd does not show the list of possible files/directories, but auto-completing names of executable files (e.g. typing b and hitting TAB) will show all the possible executables.
The currently selected auto-complete value is shown beside the prompt in all cases.
I assume this is something to do with it not scrolling down to show the output, but I can't find any more information.
I had the same problem, and after fiddling with the Customize group for term, it seems that setting the variable term-scroll-show-maximum-output to a non-nil value solves this problem (at least for ansi-term).