Firefox, in its plugins and extensions gives access to browser's "chrome" (the UI stuff around the page) which makes this possible, although I am not sure of the api that you have mentioned.
Google Chrome, however, is clear about its exposure to browser's chrome to the extension and gives access only through browser actions
and page actions
. Moreover you can use HTML5 Desktop Notifications
, Context Menus
and Omnibox
to get more out of the browser.
But I am afraid you cannot access the address bar from extensions (or otherwise) which Chrome explicitly denies access to via javascript or Extensions.
At most what you can do is to point it out to user that pressing Ctrl+L
will get them to browser's address bar or if you want to redirect the user you can use window.location.href = 'https://stackoverflow.com'
.
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