If we choose poorly, their health may suffer. In my view treatments like homeopathy, which are not supported by robust evidence, will at best cost you money and do nothing, but at worst they may delay effective treatment.
Though we are fully capable of making informed decisions about our own healthcare that might involve trying treatments that are not backed by good evidence, I strongly believe that we have a duty to our pets not to experiment on them in this way.
This responsibility also applies to veterinarians. When a vet says things that are not soundly based on scientific evidence, such as "homeopathy… can help healing tremendously", I believe they are failing in that responsibility.
The New Zealand Veterinary Association has a policy on the use of "complementary and alternative treatments" that requires, among other things, that vets must say if a treatment they are advocating is not supported by evidence:
"Clients must be made aware of the likely effectiveness of a given treatment according to recognised peer-reviewed veterinary medical publications, notwithstanding the individual beliefs of the veterinarian. They must also be told the degree to which tests, treatments or remedies have been evaluated and the degree of certainty and predictability that exists about their efficacy and safety."
I sincerely hope that your vet fulfils this responsibility in their practice.
-Mark Hanna is Chair of the Society for Science Based Healthcare