Gold is only backed by the agreement of its scarcity; otherwise, it has no value (outside its physical usefulness as a material).
Fiat currency is only backed by the confidence of a nation; otherwise, it has no value.
Time is a currency that's backed by your life.
It is as real as it gets.
The scarcity of Gold on the other hand is very real. Gold is limited and while it is possible to synthesize gold, the cost of doing that is great and does not scale.
Which is why I think the world you mention is impossible: we cannot even stop the piracy of music. Artificial scarcity is not sustainable. Once all the people will be able to live forever, they will live forever. What makes you think these "clocks" will be unbreakable?
And even if the technology will be perfect, do you really want to have on your hands a huge angry mob of people with less than 24 hours to live? Good luck with that ;-)