The Torah tells us Vayikra (19:26)
לא תנחשו ולא תעוננו
which means that a Jew is not permitted to participate in (loosely translated) omens and superstitions. We normally associate that with things like the red “kaballah” string which some Poskim rule, based on a Tosefta, is forbidden to be worn as a Torah prohibition of the above.
Perhaps more interestingly, poskim such as R’ Hershel Schachter contend that not following evidence-based (scientific) medicinal practices and treatments also contravenes this biblical command. They claim this is why the Rambam (and Magen Avraham and others) omitted the prohibition of easting fish and meat all together. If Science (peer reviewed evidence based research) indicates that something does not pose a health danger, then it is prohibited to adopt a former practice that relies on faulty medicine of the time.
It could be argued that homeopathy and its cognate areas, elements of whose treatments have not been scientifically evaluated for efficacy, pose a similar Torah prohibition. That is not to imply that all alternate medicine (now also known as complementary medicine) falls under this prohibitive rubric. Rather, unless there is a known, scientific, evidence-based peer-reviewed study which shows that a homeopathic or alternate approach is indicated as a complementary approach to a medical condition, it could be cogently argued that it is forbidden to use these alternate approaches.
RMIT has a research group in Complementary Medicine. I don’t think Halacha has any problem with that, nor would it have a problem with going to a regular doctor who is also well-read and skilled with alternative, evidence-based, scientifically-sound, complementary alternative approaches to medical conditions.