The EU did a project on hydrogen powered aircraft, which reported in 2003. Advances in materials science may make this more of a runner as we go forward. A hydrogen powered aircraft would, in principle, have 0 carbon emissions.
There is a lot of work going on on carbon sequestration, and removing CO2 from the exhaust stream is viable for some applications, such as large power plants. The kit is heavy, and not cheap to do.
I cant see carbon capture at source from aircraft working, due to weight problems, and some for want of a better word, thermodynamic considerations.
Solar powered static capture from the atmosphere machines are available, we call them trees. The political will and the space to deploy them are problematic.