Moving search field into tab

I want to move the "box" that is usually shown when you click in the search field, to be a tab similar to how you have the visitor tab with all the account stuff.
Just to clarify, it's this search "box" I want to be in it's own tab named "Search" that shows the content of the box when mouse-over, or if clicked sends you to the search page.

I've been trying to achieve this myself over the last few days, but can't for the life of me get it to work.

I would be greatful for anyone willing to help.
You can see how far I've gotten at I'm guessing it's an issue with how the search "box" is styled, though I'm still trying to figure everything out on what to change in the styling to make it work without ruining the functionality of the box

I guess what I need is help with the fixing of the styling, I'm not the best at that
Gotten some success so far:
I had to remove the common search menu from the form because it was the one messing up the whole dropdown (not a big loss in my honest opinion)
However, I can't figure out how to disable that it hides the form fields below the main search field if you click outside it, see example below:
That is what happens when you click on the border in the popup to make the search field loose focus, I'm guessing it's javascript, but I'm a bit weary to mess with the javascript unless I know what I'm doing (and I don't).

So, anyone who can tell me how to disable that hide function of the quicksearch fieldset?