The most sensible minimal extension (to 2d) that I know of is to distinguish between how much you value personal freedom on one axis and economic freedom on the other. That still leaves out a lot, but is the best indicator of which side you'll likely to end up on regarding most political contentions, given the simplicity of the model.
There are actually two popular models of that kind, the Nolan Chart (example quiz [1], your spectrum is the vertical axis) and the Political Compass (example quiz [2], your spectrum is the secondary diagonal).
Also, btw, the fascist movement recruited much of it's ideological and political concepts and initial core activist base from syndicalist/corporatist socialist circles and was strongly anti-capitalist/pro-socialist and anti-conservative. Benito Mussolini was for a time chief editor of the Italian Socialist Party's newspaper "Avanti!" and member of the party's National Directorate.