Shakespeare wrote in the English of his day, which is close to the English of the King James version (KJV) of the Bible. Though the word “methought” is not found in the Bible, it is a perfectly valid English expression given that Joseph Smith deliberately imitated KJV English in his translation of the Book of Mormon.
It is interesting that both passages in which the word “methought” appears deal with Lehi’s vision, suggesting that Alma may have borrowed the verbiage from the accunt of Lehi.
For another criticism regarding borrowing from Shakespeare, see 2 Nephi 1:14. A Quote from Shakespeare?
For a discussion of the use of KJV-style English in the Book of Mormon, see KJV Language of .the Book of Mormon