Air transport is probably stuck with liquid fuels for the foreseeable future. The energy density of batteries is way too low, and fuel cells probably don't have enough efficiency over turboprop engines to make up for hauling the weight of the fuel cell around (nevermind the extra capital cost of a fuel-cell equipped plane). You're obviously stuck with a prop plane in either case, so flight times would increase--which means crew costs also increase and flights are that much less competitive with ground transportation.

As SeriouslyLost said, a jet biofuel is a likely long-term option. Shorter term, we would probably see more synthetic Jet-A made from natural gas (aka gas-to-liquid, or GTL) if oil prices rose substantially. Numerous airlines have trialed biofuels or GTL fuels. I assume they're crossing their fingers that the prices for such fuels comes down over the next few decades.