Most of us don't have control over the public keys we use. We connect to a web site with a browser, and the server supplies the public key.
If you want public keys for your own purposes, I guess nothing stops you from generating 4096 bit (or whatever) ECC keys. Of course the arithmetic will be slow.
If you're really paranoid about potential math advances affecting ECC and/or RSA, then you may be best off avoiding public key schemes altogether. Just use secret keys, and maybe Merkle tree signatures.