I do a large amount of Google trainings and being able to snap and/or record the Omnibox is nearly a requirement for me. I do hope this gets added. As without it, this extension is not as useful to me as I would need it to be.
Thanks for the idea! I went ahead and changed your title to include capturing all things outside of the browser tab, not just the omnibox/search bar. We are considering this, so anyone else who sees this need, please check the "Me too!" button. :)
Also, bonus points for any specific use cases, like the one Karl provided here.