It would not be so much a network split as a network copy. Every user of the original network would have, to begin with, identical balances in both. Until inflation adjusted things, there would be an ephemeral increase in purchasing power.

If the new network implemented inflationary logic, as a modern state is wont to do, I can't exactly see people rushing to convert their old (relatively stable, perhaps deflationary) money to the new one. Therefore, I can't see many businesses stopping to accept the old bitcoin.

And after whatever adjustments are made, it will be business as usual. Bitcoin will remain the network for people that want a decentralized currency. Not necessarily of libertarians, but of anyone with distrust or disillusion with their governments and the banking system. Which I don't think is such a small demographic.