Writers that absolutely had to monkey with core elements could monkey in the modern era... but the original would live on in his mythic and iconic past. Each era would be a joy to write.
Note that writers ranging from Alan Moore (Supreme, Tom Strong) to John Byrne (Generations) have done this sort of thing to Superman archetypes.
Superman isn't consistently immortal. The classic LSH tales don't have a 30th century Superman in them. Lots of imaginary future tales showed him as dead or looking / acting his age when projected into the future. I kinda like that. Kurt's Superman: SI grew old and wasn't seemingly going to live forever, and that was refreshing. To me, Superman being immortal and _knowing_ that he'll be immortal is a really big deal as far as how he relates to everyone around him. Some examples:
- Do you think that a Superman who knows he's immortal gets himself so attached to mortals that he totally flips out when they inevitably die?
- Is he thinking about the "death"/long-term future of Clark Kent, knowing that an 80-year old Clark who looks in his 20s would be suspicious?
- Does he interact in interesting ways with other immortals, good and bad (think Highlander TV series)?
- How bored might he get living life for "forever" at super-speed, or how does his brain work to overcome that? (That's an issue for speedsters in general but at least most of them age, unless the Speed Force grants eternal middle-age for everyone.)
- If you know that your Super-children will live forever, would you raise them differently... prepare them for loss and ennui better, perhaps?
- What grand things would he want to do a few thousand years from now once civilization has hopefully evolved to not need him on patrol?
etc.
How much of this fits in a popular comic book is questionable? Superman being approached by some immortal who's seen it all and wants powerful Superman to help him in assisted suicide might get the riot squad out. I doubt that'd have been an issue for Alan Moore's Supreme.