“You better watch out”
In “The Greek Interpreter”, Conan Doyle wrote:
“There was a colored gas-lamp inside which was turned so low that I could see little save that the hall was of some size and hung with pictures. In the dim light I could make out that the person who had opened the door was a small, mean-looking, middle-aged man with rounded shoulders. As he turned towards us the glint of the light showed me that he was wearing glasses.
“ ‘Is this Mr. Melas, Harold?’ said he.
“ ‘Well done, well done! No ill-will, Mr. Melas, I hope, but we could not get on without you. If you deal fair with us you’ll not regret it, but if you try any tricks, God help you!’ He spoke in a nervous, jerky fashion, and with little giggling laughs in be- tween, but somehow he impressed me with fear more than the other.
“ ‘What do you want with me?’ I asked.
“ ‘Only to ask a few questions of a Greek gentleman who is visiting us, and to let us have the answers. But say no more than you are told to say, or—’ here came the nervous giggle again—‘you had better never have been born.’
“You better not cry”
Saying you had better means it would be best for you. Merriam-Webster says it means “would be wise to”, and Collins adds that it (as well as you had best) is used to convey obligation or compulsion, as in ought to.
Oxford Dictionaries Online (not the OED), says that it means “Would find it wiser to do something; ought to do something”, and offers for had best do something “Find it most sensible or well advised to do the thing mentioned”.
“He’s making a list”
These all take infinitive complements. Use the straight infinitive to convey the present:
- You should be on your best behavior.
- You ought to be on your best behavior.
- You had better be on your best behavior.
“And checking it twice”
And use the perfect infinitive to convey the past:
- You should have been on your best behavior.
- You ought to have been on your best behavior.
- You had better have been on your best behavior.
“Santa Claus is coming to town”
In a 2013 Christmas article on Six grammar points to watch out for in Christmas songs, Arika Okrent wrote in The Week:
Though the "had better" construction has been a part of English for 1000 years, it came from a distortion of phrases like "him were better that he never were born," where "were" was a subjunctive ("it would have been better") and "him" (or "me," "you," "us") was in the dative case ("him were better" = "it would have been better for him"). People started changing the dative to the subject case ("he were better") and then changed the "were" to "had."
That was all hundreds of years ago. Then, in the 1800s, people started dropping the "had." The grammar books of the late 1800s tried mightily to shore up the "had" (some even making up a rule from nowhere that it should be "would," as in "he would better"), but these days the bare form is considered correct, if a bit casual for formal contexts. Clearly, "Santa Claus is Coming to Town" wants nothing to do with fancy formality. So "you better watch out" is the way to go.
So the phrase by Conan Doyle is just fine. It is not a “past perfect” form at all. Think of it as a modal use, one preterite in form and like other modals taking an infinitive for the present and a perfect infinitive for the past.