At the very least it is useful to split Social and Economic classification. The natural correlation between the two is far looser now it has been historically. Take Cameron, economically a mentalist, but very liberal for a Tory in social terms. Similar some of the left's dinosaurs are socially illiberal.
Indeed, and why the traditional one-dimensional continuum should perhaps be considered a gross oversimplification in this day & age, rather than a) loosely fitting the political terms 'left' & 'right' to your contemporary meaning, more or less just making them bywords for collectivist/statist liberalism & anarcho-capitalist quasi-fascism, or b) tightly squeezing yourself into a crude paradigm you don't actually comfortably fit, just to belong somewhere in the 'mainstream' political landscape (or like someone with certain conscientious socialist leanings feeling a pressure to subscribe to a quite hardcore Socialist party, then toeing the party line on issues they don't internally harmonise with, and gradually having their own political sensibilities dragged that way over time).
The two-dimensional political compass method is probably the most elegant and comprehensive way of pinpointing your political place on the map, I reckon, while still being very easy to grasp, with its natural 'cardinal points' giving you a nice picture of all the ideological space around you, and where the extremes lie socially and economically, without conflating distinct orientations just for the sake of convenience. It does require far more in-depth - and less biased - questionnaires to plot it accurately than has likely ever actually been used for that purpose, though.
A 3D chart is probably the ultimate model, with the third axis being some sort of 'size of state/people power' measure to go with the economic and social axes - from statism to anarchism, I suppose. Being 'far left' economically could mean favouring an omniscient, omnipotent soviet union-style "workers' republic" ideal state, or a loose collectivist new-age tribal commune thing; economically 'far right' can mean an unregulated free market pumping up the all-powerful state, or pretty much every man/gangster [in it] for himself. Socially far left can mean a massive central "nanny state" looking to provide for everyone whoever you are, or a completely devolved liberalist regional autonomy thing; socially far right could be a strong authoritarian state with staunchly conservative values looking after
only their own, or an illiberal mob left to their own power struggles.
Whether such an added axis would serve to 'complete' the picture to all intents and purposes, I dunno, but three dimensions may already be going into too complex a territory for many, without needing to complicate things yet further. Also, not everyone in the world sees democracy as the sacred foundation for everything; some would prefer a 'benign dictator', some a 'meritocracy', and so on. Although you can't plot all that crap on a single axis, a line running from state power through to people power (or more accurately, ultimately, 'person power') can at least suggest some more detailed governance stuff.
X = Economy: Collectivism - Individualism (Anarcho-Capitalism)
Y = Civil Rights: Liberalism - Authoritarianism (Fascism)
Z = Power Structure: Statism - Anarchism (Libertarianism somewhere around there)
You can get an authoritarian centralised capitalist state, and decentralised authoritarian Wild West-style law-of-the-jungle rabbles. A very liberal & all-encompassing tax-collecting nurturing welfare state, and loose-knit decentralised liberal communities looking after each other.
Anyway, I guess I'm sort of quite left-of-centre on X (socialist leanings, but totally cool with reasonably regulated private enterprise, etc.), very low on Y (as in high on the granting of personal freedoms), and fairly central-deep on Z (leaning more toward decentralised government, but not going mad with it, however tempting that is).