Just regarding TM- I had to do some research for it for Debate(very long story), and some of the information I found was this:
TM doesn't work for everybody. People complain of symptoms from practicing ranging from headaches to seizures to general emotional weirdness.
The important part of TM is not repeating a given mantra- the important part is the stillness and general relaxation. They did a study where they had a bunch of people do TM, and a bunch of people were told that they should think about as many things as possible at one time, and a bunch were told to just sit still. In the end, all groups reported the same positive(and IIRC, negative) phenomenon.
TM won't necessarily make you fly :-). There are actually several lawsuits regarding this in the courts...some people received upwards of $100k.
Hey, I'm not advocating TM, but my adaptation happens to work for me. I find that focusing on a phrase works to still my very active mind.
As to the flying stuff, I had no idea. When did they start that? I looked into this stuff around 20 years ago so maybe they went off the deep end in the interim? Presumably it's like yogic flying? Bounce! Bounce! Bounce! And if it wasn't for gravity...