Grammatically the first three are correct, and are essentially the same thing, but with "have" implied instead of directly given in the first and third. The fourth is grammatically incorrect, because one should not say "the same hobby as me have", so the same is true when merely implying the verb instead of explicitly giving it. "...the same hobby as I (have/do)" - correct "...the same hobby as me (have/do)" - incorrect

I have heard them all, I would say in almost equal amounts (measured in the number of times I have heard them). My advice would be to check with your teacher if you are studying for a specific English Exam. To verify what the examiners are expecting what they might mark down and what they would consider correct. If working for a newspaper etc then stick with their "style". Number 3 is what I believe to be the correct answer to fulfil both of these options, and sounds correct normal and natural to my ears.

They are all arguably 'correct'. 1. He has the same hobby as I. = Grammatically correct, but awkward. 2. He has the same hobby as I have. = Unusual, but as Aisling says, probably an acceptable regional variant 3. He has the same hobby as I do. = Fine. The 'best' answer, in that it is the least controversial. 4. He has the same hobby as me. = Purists would argue that that object case pronoun ( 'me') is wrong, but it follows a pattern that is very widely used among native speakers. It's normal English - I don't have a problem with it. If you're doing a written grammar test - choose number 3. In order of 'normalness', I'd order them 3, 4, 2, 1.

I think 4 is correct

2,3 and 4 are correct for me (I'm Irish), but I think a lot of native speakers would consider 2 to be wrong- (Irish people use have in questions and negatives without an auxiliary verb and we say things like "I haven't any time", which other native speakers find strange)