Short answer: yes, hydrogen jets are probably the future. Electrochemical batteries fundamentally lack the required mass to energy ratio necessary for air travel. But there's a lot of challenges to actually powering an aircraft via hydrogen, and it's probably not going to happen for a while.
The long answer:
* The question is moot until we have widespread decarbonization of the energy grid. Currently, our hydrogen mostly comes from steam reformation - splitting methane, CH4, into 2H2 + CO - which emits carbon dioxide. Once our electrical production is carbon-free, then we can think of producing hydrogen and powering vehicles with that. Electrolysis can serve as a carbon-free source of hydrogen, if the electricity is produced from a carbon-free source. There are more efficient forms of hydrogen production, like thermochemical hydrogen production with heat provided by nuclear reactors.
* Once carbon-free hydrogen production becomes available, then there's the question of building hydrogen powered vehicles. Hydrogen is the most energy dense fuel by unit of mass, except for nuclear fuels, but it's about half the energy density by unit of volume compared to hydrocarbons. Substituting fossil fuels with hydrogen is probably more feasible in container ships than aircraft. Hydrogen containment, be it pressurized or liquid-cryogenic, scales better at higher capacities as there is a better volume to surface-area ratio. Aircraft wings (long and skinny) are basically the worst shape for hydrogen containment. Past examples of hydrogen powered aircraft stored it in the fuselage [1], which would sacrifice cargo and passenger space.
* Lastly, for aircraft there's also the question of making engines that don't produce greenhouse gases. High-temperature hydrogen gas turbines will also produce nitrous oxides, which are greenhouse gases. This can be mitigated by running turbines at lower temperatures, but that reduces efficiency. There's also the idea of using hydrogen fuel cells to produce electricity to run turbines, but fuel cells and electrical motors don't have as good power to weight ratios as combustion turbines.
I'm much more optimistic about converting maritime transport to hydrogen fuels than aircraft. That said, the military is very interested in hydrogen planes. One of the limiting factors in modern naval warfare is fuel to fly planes. With hydrogen powered planes, a carrier could produce hydrogen with its nuclear reactors giving carriers an effectively unlimited source of aviation fuel. If they figure it out, hopefully the tech propagates to civil air transport.
1. https://en.wikipedia.org/wiki/Tupolev_Tu-155