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Using “better” by itself is fine except in formal English. “In a wide range of informal circumstances (but never in formal contexts) the had
can be dispensed with,” Fowler’s says.
Merriam-Webster’s Dictionary of English Usage calls “had better” a standard English idiom and agrees with Fowler’s that “better,” when used alone in this sense, “is not found in very formal surroundings.”
The Oxford English Dictionary’s earliest citation for the construction without “had” is from a pseudonymous letter to a newspaper by “Major Jack Downing”:
“My clothes had got so shabby, I thought I better hire out a few days and get slicked up a little.” (The letter was published in a book in 1834 but was written in 1831.)
The OED says the abbreviated usage originated in the US, and labels it a colloquialism. But Merriam-Webster’s Collegiate Dictionary (11th ed.) lists it without reservations.
The Merriam-Webster’s editors give the example “you better hurry,” and says “better” in this sense is a “verbal auxiliary.”
It should be noted that even the full phrase, “had better,” was criticized by some in the 19th century on the ground that it was illogical and couldn’t be parsed: an 1897 issue of the Ohio Educational Monthly says many teachers found “had better” and other idioms “very difficult to dispose of grammatically.”