I want to be able to see today, overdue, p1, p2. Is there a way to save this query so it appears as an option in the "more" menu (or somewhere else)?
Via the preferences screen, it is possible to edit the search conditions for your start screen. The start page appears as one of the options in the menu of possible queries.
unfortunately, there is no particular function allowing saving queries. However, as Martin mentioned, you can go to Preferences -> General and choose "custom query" in the "Start page" drop-down menu.
Also, please note that if you create a bookmark (or link) to a query result page, you'll be able to access the search result directly.
You can vote for this at https://todoist.com/Vote/showProposal/380/