I invented such a theorem in the late 1980s. Patrick Morton told me I should publish it, but I thought it was too trivial. Let F(z) be an analytic function. My idea was that there is some curve A in the complex plane, on which Re(F)=0. There is another curve B on which Im(F)=0. If you know that for some domain D in the complex plane, topologically a disk (F analytic everywhere inside), that A and B cross the boundary of the disk in the cyclic order along that boundary ABAB, then we are topologically forced to have the curve A crossing the curve B somewhere within D. And at that crossing point, F=0. The curves A and B generically cross each other orthogonally, and neither can be self-tangent due to conformal properties of analytic functions. In theorem above, you need to assume that those 4 points of the boundary of D must be unique and that the curves actually cross the boundary there, e.g. the right one of Re(F)' and Im(F)' where here ' represents derivative along boundary, each is nonzero at the 4 points (without these, theorem inapplicable). If you do not want to assume F is analytic, well, mere continuity suffices for some but not all parts of the argument... --- One can also obtain complex versions of Rolle's theorem...