Sergio, welcome to the forums.
It’s definitely an interesting idea, but I feel like having two search boxes on a page is not best user experience - we already do have one search box on almost all index pages. We do have a long-term goal providing single search box in the future however, that would be appreciated if you would like to work towards that. Be warned this is ton of work.
But let me ask you, why would you want to do that? Is my assumption correct that you find our main menu confusing? I find it too and some of us agreed on this, yet we were not able to come up with some good solution to this problem:
I belive the menu needs to be redesigned.