Hodges claims that Turing at least had some interest in telepathy and prophesies:

These disturbing phenomena seem to deny all our usual scientific ideas. How we should like to discredit them! Unfortunately the statistical evidence, at least for telepathy, is overwhelming. It is very difficult to rearrange one’s ideas so as to fit these new facts in. Once one has accepted them it does not seem a very big step to believe in ghosts and bogies. The idea that our bodies move simply according to the known laws of physics, together with some others not yet discovered but somewhat similar, would be the first to go.

Readers might well have wondered whether he[Turing] really believed the evidence to be ‘overwhelming’, or whether this was a rather arch joke. In fact he was certainly impressed at the time by J .B. Rhine’s claims to have experimental proof of extra-sensory perception. It might have reflected his interest in dreams and prophecies and coincidences, but certainly was a case where for him, open-mindedness had to come before anything else; what was so had to come before what it was convenient to think. On the other hand, he could not make light, as less well-informed people could, of the inconsistency of these ideas with the principles of causality embodied in the existing ‘laws of physics’, and so well attested by experiment.

Alan Turing: The Enigma (Chapter 7)

I think Turing's willingness to take all comers seriously is something to emulate.