What would Bitcoin be like with two block chains? I once heard somebody say that to overcome the absolute supply limit of 21 million bitcoins, one could start their own block chain.

Besides the adoption concerns (most likely nobody would accept the new block chain as genuine in the currently existing Bitcoin network even if it had its own genesis block and wasn't just a fork), are there any networking problems with this? If, hypothetically, the adoption concerns could all be overridden, would there be any data conflicts with the existing block chain? Is the Bitcoin protocol designed to be able to use more than one block chain? Can the client handle it?
Most other crypto coins is more or less what you describe.

The bitcoind code has build-in "checkpoints" (block height x has hash y), but a new blockchain would anyway use a "new" ( / forked ) client, so you could just remove these checkpoints.

However the 21 million bitcoin limit will never be a problem. Today each bitcoin can be divided down to 8 decimal places (0.00000001 BTC), should we for some weird reason need more bitcoins, then this limit can be changed.