I wrote a paper last year that began to discuss that very matter. It also dealt with how Superman's story is allegorical to the story of Christ, discussed how his story corresponds to Joseph Campbell's heroic pattern, and discussed how Superman in general appeals to people's subconscious psychological desires. It was 25 pages long (double space) so obviously I did not have the room necessary to truly do the thorough job with it that I some day hope to do.
As much as I admire Campbell's work, I think an analysis of Superman exclusively through Campbell would be incomplete because it would look at the story in terms of psychology and Jung and symbology, and so forth, while a character like Superman is the product of several writers working at a specific time, place, medium, and situation, and consequently looking at the time, place, situation, and medium may be more accountable for explaining why Superman is the way he is.
The danger of a "Hero's Journey" essay is that it makes analyzing works out to be a sort of Easter Egg hunt, cherry-picking details to fit the pattern. Campbell creates a formula, and with archetypes that are by definition vague, all you have to do is plug new numbers into it.