A snippet from the Grammarphobia blog post, “You better believe it”
Q: I’m Australian, and an American friend often says things like “I better not forget it” instead of “I’d better not forget it.” Is this correct? Is it a case of US usage differing from UK/Australian usage?
A: The idiomatic phrase “had better” (as in “I had better study” or “We’d better go”) is a venerable usage with roots far back in Old English.
The shortened form “better” (as in “I better study” or “We better go”) dates from the 1830s and is used informally in both British and American English.
In fact, Fowler’s Modern English Usage (rev. 3rd ed.) says it’s not unheard of in your neck of the woods: “In practice this use of an unsupported better is much more common in North America, Australia, and NZ than in Britain.”
Using “better” by itself is fine except in formal English. “In a wide range of informal circumstances (but never in formal contexts) the had or ’d 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.”