One of the main reasons quantum physics has become the mystics' icon is that many notable physicists melded it with mystical ideas. But why quantum physics?
Quantum physics is a highly abstract theory with characteristics that outperform the weirdest phenomena introduced by relativity theory several years earlier. One example is tunneling, whereby a quantum particle located on one side of a solid (quantum) wall can be found on the other side. Other, considerably more mysterious than tunneling, are the notions of non-locality and entanglement. These are hopelessly inexplicable in ordinary languages. Their very suggestion required years of attempting to digest quantum physics.
Non-locality, for example, was the result of Einstein's desire to prove the incompleteness of quantum physics. He and two of his colleagues wrote a paper in 1935, almost a decade after Schrödinger's equation, in which they demonstrated the "spooky" non-local behavior of quantum physics. The fact that it took almost thirty years to prove -- mathematically, and only mathematically -- that quantum physics is non-local and there is nothing wrong with that, illustrates the degree to which such phenomena defy explanation.
The founders of quantum physics, however, could not swallow the weirdness of their own creation. They were looking for terrains of knowledge beyond mathematics and physics that could "make sense" of the strange behavior of quantum physics. This wasn't unlike Newton's "making sense" of the motion of the planets by having God place them at the right position, give them the initial push in the right direction, and let the mathematical law of gravity and the second law of motion take over subsequently. But "God [did] not play dice" Einstein famously said about quantum probability. Besides, in the intellectual circles of the mid 1920s, God had already been killed by Nietzsche.
On the other hand, Buddhism and Hinduism -- which were gaining popularity in the intellectual circles of Europe as a result of movements like Theosophy and Spiritualism -- did not have an almighty God who was the cause and determiner of everything. As the indeterministic nature of quantum physics seemed to assent to the free will, which the deterministic Newtonian physics appeared to dismiss, the founders' philosophy of choice became Eastern theosophy and their philosopher of choice Arthur Schopenhauer, whose emphasis on mindless, aimless Will and its influence on reality through the process of objectification "made sense" of the probabilistic quantum physics.
Once the founders mystified quantum physics, no one, not even Einstein and Planck could unmystify it. And some quantum physicists in the generations coming after the founders kept the mystification alive.
The baselessness of a mystified quantum physics is manifested in the admission of the mystic physicists themselves.