Other than flora and fauna, I'm not sure that humankind has actually developed a net negative carbon capture device. Have we?
If we have, I'd be super duper excited to see some links to examples.
If you want something more aggressive, I would explore growing high surface area algae in a fish tank. You can probably even make it look nice.
If I was going to homebrew something I would try encouraging algae growth on aquarium bio balls, probably with a nice waterfall for circulation.
If you could encourage algae growth on even something like this it could get you started..