It's certainly doable but it's a rather bad idea. If you're switching from one page to another AJAX kind of loses it's purpose. The idea is to be able to switch or update certain elements of a page without having to make the whole round trip to the server but if you're loading a totally new page then that round trip is justified.
What you could do, if you're really really sure that you want to do this is to put an index() method in your controller. In that index method you check if Director::is_ajax() and if it is you just render the Layout template and not the rest. If it's not an AJAX request then you render all of it like you usually do.
I think that's exactly what I want to do. In this case, the page type is the same so all of the scripts, css-files and so on can be the same, only the content-box would have to be loaded. I'll try the index-function you mentioned. Thanks.