You find both accusative pronouns (me/him/her/them) and nominative pronouns (I/he/she/they) in this syntactic position in standard English. The forms with the nominal genitive pronouns (mine/yours/hers etc.) are a red herring because they stand for something possessed rather than the person themself.
The traditional rule for comparison with a person is that you must use nominative. However, according to my research, accusative is more common.
I searched the Corpus of Contemporary American English for this syntactic structure, followed by a comma or a period to ensure we are not looking for cases like faster than he is, with a verb following the pronoun, in which case nominative is obligatory.
There were 1046 results for the accusative pronouns and 450 for nominative pronouns, more than 2 to 1 in favor of accusative pronouns—the “traditionally wrong” form. Both forms are standard, so my advice to a writer choosing between these forms is to consider that the “traditionally correct” form is unimpeachably correct but a bit formal. Choose the form that best matches tone and formality level of your writing.
For the curious, the queries looked like this:
[jjr*] than me|him|her|us|them .|,
[jjr*] than I|he|she|we|they .|,
where[jjr*]
means any comparative adjective.
Update 2011-05-23
Using the new Google Book Corpus search, I was able to construct a Google ngrams-like graph comparing these usages over time, using these two queries: accusative, nominative:
As you can see, until the late 1980s, the formal usage was more common than the informal usage. Since then, however, accusative has very rapidly eclipsed nominative, even in this corpus, which represents professionally published works.
You could say 'I'm waiting two hours', and no one would misunderstand you. I don't think there is any loss of meaning, but you might sound a little foreign. I think this usage is standard in some types of American-English. I could imagine a character on Seinfeld going 'Two hours I'm waiting - what is it with these people?’
In speech (but not in formal writing) you can say 'How long've you...'
Actually, there is a context where "I am waiting two hours" means something completely different (thanks Claudiu, Kosmonaut). If the issue was for example how long you were prepared to wait for a delayed flight, then you would say 'I'm waiting two hours (then I'm going home!)" to mean "I will wait two hours only"
So if the context is unclear, you need the 'have been waiting' construction.
Best Answer
Your first sentence uses a present perfect progressive active construction, while the second is a passive construction. Each has a different emphasis.
The second doesn’t necessarily mean that the computer is no longer running, but the first would in any case be more normal. You could make it sound less formal by writing For How long has your computer been running for?
If you know that the computer has stopped running, you could ask How long was your computer running for before it stopped?