I have never seen any verb form following modal verbs other than a bare infinitive.
The bare infinitive of to have is have. You should always use have after could. There are many typos on the internet, i.e., could + had+ past participle and you should compare it with could + have + past participle.
If you Google "could had" now, you get 395,000 hits, but many of them in the first page are questions about whether could + had is correct or not.
If you Google "could have" now, you get 323 million results.
Sometimes, people write could + of + past participle, but it is also a mistake.
[Oxford Online Dictionary]