"I am older than he "

"I am older than he is." This is formal and correct. The other one [I am older than he], is also correct, but is seen in literature, Shakespeare.

"I am older than him" is correct. You need a object pronoun here, not a subject one. To use "I am older than he" would be incorrect. It's like when people say (thinking they are speaking high quality English) something like "Bob and I are friends". This is wrong. It should be "Bob and me are friends".

I was taught they are ALL correct. But I might be very wrong about it though. The fact is "than" could be a preposition, or also a conjunction. Hands down. The "than he is" one is in the complete form, no words are omitted. Therefore "than he" is a correct alternative that is short for "than he is", where "than" is both used as a conjunction, like the usage of the word "and", which can only be used as a conjunction, "I am her friend, and I like her (followed by an entire sentence)", "I like blue, red and green (followed by a sole noun)". "Than him", where "than" is as a preposition, is just like "I do it totally FOR him", "I believe IN her", "he cheated ON me", "Let me sing a song TO/TOWARDS/in front OF her". Rather "than he is" is just not as commonly seen as "than he/him". And I assume "than him" is the most frequent choice of most people and seems to be more natural to their ears. Similar situations also refer to words like "except", "but", "as", "like (informally as a conjunction)", "after", "before", and so forth. Very good Q.

'I am older than he is.'