There was some discussion of this back in 2013, but unless there is something wrong with my site configuration it doesn't look like it has moved forward. Setting a profile picture would be a great criteria for profile completion.
Any way to do this?
Previous discussion: https://moodle.org/mod/forum/discuss.php?d=226938
The criteria for awarding badged haven't changed, so the simple answer is no
As Simon said in the original post, adding criteria is a straightforward task for a developer.