In our projects we would like to include as little jQuery as possible. Therefore we'd like to bypass the Ajax framework included in OctoberCMS. However, it seems that the framework is rooted pretty deeply in the system. Would it be possible to easily bypass the framework? And what would be the best way to get started?

Already figured it out. You just have to set the correct request headers. Then you can return data from the controller like so for example:

return response()->json($this->groupResults($results));