Dr. Luc Montagnier, the French virologist who won the Nobel Prize in 2008 for discovering the AIDS virus, and Brian Josephson, Ph.D., who won a physics Nobel in 1973, are both strong proponents of homeopathy.
This is the idea that you add to water some chemical you think is hurting you, dilute it several times, it becomes a tonic! Say, add some cyanide to a liter of water, mix, then take out a centiliter of the mixture, and add to a liter of pure water; take out a centiliter of the mixture, etc. You can do this 20 times and basically no salt is left. Homeopathy believes that water has a 'memory' of compounds that were once dissolved in it, which through another mysterious mechanism explains how homeopathy works. Skeptics of homeopathy, like me, think homeopathic medicines have nothing in them because they are diluted too much. Poison and medicine is a function of dose, after all.
I like this because it highlights people who are experts are not experts on everything, even within their field, and they often have insanely stupid ideas. That Heidegger was a Nazi, or PCR inventor Kary Mullis believes in astrology, John Nash was a paranoid schizophrenic, Newton believed in Biblical numerology, are not a paradoxes: the most imaginative people are the most credulous because they are prone to dismiss common sense. For creative thought, common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones. In other words it can only act by suppressing originality. Original minds find great unconventional ideas, but also lots of bad ones.
We all are reminded not to stereotype groups of people, but too often are told if someone has a wacky statement or belief, or does not understand something you think is obvious, this disqualifies them from debate on important issues (Paul Erdos couldn't understand the Monty Hall problem). Richard Dawkins commented on a case of religous descrimination at the University of Kentucky, arguing that if one believes in God that should be relevant to their jobs even if they keep such beliefs private, or if it such beliefs had no obvious relation to their job (say, an eye surgeon who is otherwise competent believing in fairies). He basically thinks one can't be really good at a job if you believe in things he finds stupid. I obviously disagree--not just with what he thinks are stupid ideas, but also that such stupidity disqualifies one for applying idea in other areas. I find comfort in the many perverse beliefs in creative minds, perhaps because I disagree with the consensus on many things and it's self-serving.
In Brian Greene's new book Hidden Reality he promotes the idea of the multiverse, one of the oldest is the many-worlds theory, which conjectures that all of the possible histories of our world allowed for by quantum mechanics are realized in other universes. So, there's you winning the lottery, you as a rock-star, in all these other worlds. It's an intriguing idea, mostly because its the only one I know of that can explain the EPR paradox, but it's totally untestable, with no implications other than explaining reality via a fanciful reality. It is like the concept of God, it explains everything using faith.
Yet, I'm sure Dawkins would consider such physical musings reasonable. Indeed, Lawrence Krauss, who is often featured with Dawkins lambasting religion, spends his time working on a theory that explains the what was going on before the big bang, as if that was real science. The distinction between what Krauss finds so objectionable--religion--and his day job, is to me a matter of semantics.
I'm profoundly against most arguments I come across, so it's not like my mind is totally open. One of the arguments I really don't like is that if someone believes something I disagree with he must be an idiot, or that if he's a genius in number theory his opinion on global warming is obviously rigorous and true. People have multiple selves, as some people who are warm to their family are monsters to their colleagues or vice versa. Most importantly, a person with only 'reasonable' ideas would probably be a good manager, but they wouldn't be really creative, or interesting.