I am getting the same AJAX errors being reported by others, either when adding a field or saving the form. I just spent several hours to see which version of wordpress or which plugin is causing the problem but I kept running in circles. I went back to my original installation of everything and I now see that if I click an action multiple times, it works once after perhaps several failures. So I guess I have a poor workaround for now.
Any suggest on how to track down the problem?
I have already: increased memory to 128M, uninstalled everything and re-installed, and activated plugins one at a time. The minimal config to get an error seems to be wordpress 3.4.2 and nextgen-gallery which was the first plugin I activated. Once I see an error, the problem seems to be "sticky". I can deactivate, and delete the other plugins, and reboot the server, and the problem continues. While the errors are happening, I can install a second website instance on the same server and it works ok. I have debug logs etc, and I don't see an error message logged. When it happens, the access log does not record a post to admin-ajax.php.