I would like to delete AJDT completions from CTRL+Space menu. I had AspectJ Tools installed, but I removed the plugin, but the suggestions remain.
Are you sure that AJDT is uninstalled? Go to Help -> About Eclipse... -> Installation Details -> Features. Look for any org.eclipse.ajdt.* feature. Now go to the plugins tab and look for any org.eclipse.ajdt.* plugins. I would bet that some remain.
If so, go to the Installed Software tab and look for org.eclipse.ajdt.feature.group. If you don't see it at the top level, then this software is transitively installed by being required by another feature or plugin.
Try that first and get back to me if you find anything. There are several ways to force a removal, but this would potentially damage other plugins that depend on AJDT (eg- SpringIDE).