Support

The problem is that any plugin would be called from the context of an AJAX request. I can see the following problems:

• If the plugin fails for any reason or causes a timeout the backup reports an AJAX error and people will come to us for support. It will be impossible for us to convince our clients that no, we are not stupid, it's a third party plugin doing something bad.
• We are moving away from a platform-specific implementation for our backup software and into a common code base for all backup products. Having a plugin event for Joomla! would be inconsistent at best and most likely problematic for other platforms.
• The proper way to get notifications for complete backups is by using Finalization classes. There is currently no way to register arbitrary Finalization classes.

If anything, I'd like to let you do the latter. Your users would have to explicitly enable the third party Finalization classes. In this case I'd print a BIG FAT WARNING™ that third party Finalization classes are loaded and no support will be provided unless you can reproduce an issue without them loaded. Does that sound better?

You can always issue a redirect at the end of the backup. Just send the user to index.php?option=com_akeeba&view=backup&autostart=1&returnurl=BASE64_ENCODED_RETURN_URL&profileid=PROFILE_ID&token=JFORM_TOKEN where:
BASE64_ENCODED_RETURN_URL = base64_encode($url);
PROFILE_ID = (int) $profile_id;
JFORM_TOKEN = JFactory::getSession()->getToken();

This auto-starts a new backup with profile number PROFILE_ID. At the end of a (successful) backup it redirects you to $url. The URL should be internal or JApplication in the latest Joomla! versions will ignore you ;)

Regarding allowing you to hook up to the end of the backup process I just realised it would be impossible. You assumed all along that the only valid backup method is the back-end backup, i.e. a Super User or a back-end user with the akeeba.backup ACL privilege is logged in to the administrator back-end of the site. Plugins and finalization classes would also be triggered in the case of front-end backup, JSON API backup and CLI backup (since we go through JApplicationCli). Therefore allowing you to trigger plugin events / custom classes would definitely break our software. For this reason I decided to not implement any such feature. You'll have to use the special backup URL I posted above. You can see that special URL in action in the system/backuponupdate plugin ;)