In a comment, John Lawler wrote:
In America it's /supərmæn/, with the /æ/ vowel of man, Stan, hand, and the second syllable of banana. What some people call the "short A" sound. In the UK that sound does not extend quite the same way, since the TV hero called /bə'nanəmæn/ in RP would be called /bə'nænəmæn/ in the US.
And Peter Shor similarly wrote:
The /æ/ sound in American English is somewhat closer to the /ɛ/ sound than the /æ/ in British English (but not as close as the /æ/ sound in New Zealand English). This is probably what is confusing the OP. I (an American) hear the American pronunciation in the link as /supərmæn/ and not /supərmɛn/.