I know the thread is old but since it's been revived and is one of the very few I can find on the internet...
I saw the double Crescent Moon two nights in a row when I was in Spain back in July 2018. Both nights I went to bed around 8:00 pm and was up around 3:30 am. So I was the moons sometime between 3:30 and 4:30. I had no drugs or alcohol in my system, have perfect vision, do not have diabetes, had no sleep in my eye, was not looking through a window or any other glass, and do not have cataracts. The moon was still far above the horizon. There was no shimmer, no stutter, no waves, no difference at all between the two. There was one crescent moon, perfect shape and brightness, then a thin black line, then another perfect crescent moon of similar (same) size, as if one was cradling the other. There was also no wind nor was the temperature abnormal.
Now the superior mirage states the mirage appears over the object but this was not the case because I could see the faint outline of the rest of the moon and the crescent on top lined up with the curvature, while the one on the bottom would have been off, by it's position alone.
Nothing else was seen in double and the two moons lasted 30 minutes to an hour before I went back inside. Again I saw this two nights in a row. Also I'm going to have a very tough time buying into the mirage theory since: 1. I've never known anyone other than the three people I found online who ever saw this; 2. In 43 years I have never seen this mirage nor heard anyone mention they had seen it; 3. To my knowledge it was not reported in any of the local papers and I did not hear anyone else in town mention it (maybe it was a "for my eyes only mirage"

).
Just to be really clear - if either of those crescent moons had been on their own, no one would have questioned it was the moon. There was nothing skewed or abnormal about either one, they were both perfect replicas.
Am I suggesting there are two moons? I'm not suggesting anything, other than there has yet to be an explanation that fit. Happy to learn of one, and happy to post this so others don't feel alone. Thanks in advance for any ideas.