"She thought that must to try alternative therapies" - Bad structure, it should read "She thought (that) she must try alternative therapies." "She thought I should try alternative medicine". - Good, but note that now, it's the speaker (*I*) who's trying the medicine, not the *she* who is suggesting.

Maybe the second is correct

"She thought I should try an alternative medicine." "She suggested that I try an alternative medicine." "She recommended that I try an alternative medicine." All of these are correct ways to say what you want. If it's just a singular medicine, you have to have the word "an" in there. I hope this helps!