More frightening are the web pages you get searching for [natty disable application menu]. The top match has you moving shared libraries around. Several others have you setting environment variables, or "simply" doing a whole bunch of removing and add operations. This page is in the top 10 and is the only one recommended removing the indicator-appmenu package. To mlaverdiere's point, there are 6 different packages with "indicator-app" in their name plus several more related ones.
You get the app menus foisted on you even if you choose "classic" desktop. I don't think it is particularly fair doing that since no desktop ever existed with them in the past so it can't be classic!
And most importantly 'apt-get remove --purge indicator-appmenu' doesn't actually work. In classic desktop you will get the menus on the apps, but the first login gives a message asking you to delete a panel item, and the panel has a "No indicators" item permanently present you can't get rid of.
(My own use case for turning them off is that I have large screens with many windows open and use focus follows mouse. appmenu is unusable for that combination.)