Well it seems to me the Cauchy-Riemann equations say quite a lot. After all once you know the real and imaginary parts are harmonic, then you know their derivatives are as well, and $$f'(z)=u_x(z)+iv_x(z)$$ so you can do it again. I don't think this is boot-strapping it, but if anyone greater on the complex food-chain cares to disagree, I'm likely retreat quickly at their word.