Theoretically Monero would be a possible solution, but only if your sole requirement is anonymity. Heavy emphasis on
only if though, since payment systems are much, much more than that requirement (i.e. dispute handling, chargebacks, insurance, fraud identification, etc.).
I don't think there are any crypto implementations today that could compete with existing mainstream systems, which in itself is a statement that opens a can of worms on HN, but maybe someday.