I suggested he come here.
Shouldn't it be "I suggested that he come here"?
Words which introduce an embedded clause can be omitted in English. It's very common, especially in speech.
All the ingredients you need are in the cupboard.
All the ingredients that you need are in the cupboard.
Both the above sentences are correct English.
"I suggested he come here" means that you suggested to him to come here (like gave a request to come to where you are - "come here!").
"I suggested he comes here" means that you suggested (to him or someone else) that he has visited where you are more than once.
"I suggested he came here" means that you suggested (to him or someone else) that he has been, once in the past, where you are.
Source: I'm a native English speaker.
However, the latter two could also be used to mean the same as "I suggested he come here" but is probably not quite correct usage. All seem grammatically correct to me, just subtle differences in meaning.
I was wondering if this "I suggested he come here" was a left over piece of the English subjunctive that is only noticeable in the 3rd person, or if I am thinking of something else.
it is advisable that he studies vs it is advisable that he study (the difference between these)
its ok if you don't know. I don't really know but you seemed to have English grammar done pretty well. Regardless great explanation!
Yes, those 3 are all fine to native ears.
You are absolutely right about the subjunctive. It's explained in a lot of detail in https://en.wikipedia.org/wiki/English_subjunctive