Consider the polynomial f(z) = (z-1)(z-2)...(z-20). Clearly it has 20 roots; these are 1, 2, ..., 20.
Now consider the polynomial g(z) = -z20. It also has 20 roots, namely the origin with multiplicity 20.
And consider h(z) = t f(z) + (1-t) g(z), as t varies from 0 to 1. (Most of the "action" happens when t is very near 0 or 1, so this probably isn't the best parametrization.) Now, as t varies, you can find the roots numerically. Imagine the roots as twenty particles moving around in the plane. What happens, basically, is that as t increases roots start by sliding along the real axis, towards each other in pairs -- this is what you expect if you just plot f(z) as a real polynomial. (Interestingly, the collision appears to be perfectly elastic.) They then bang into each other and head off in the positive and negative imaginary directions. And eventually they curve around and approach the origin, on paths spaced 18 degrees apart. (I can try to produce graphics.)
There's nothing special about these polynomials -- that is, I suspect that something like this happens more generally. This is actually just an extension of an example in Peter Henrici's Applied and computational complex analysis (volume 1, p. 282) -- Henrici says that J. H. Wilkinson looked at the polynomial f(z) - 2-23z19 and saw that it had five pairs of complex conjugate zeroes.
But it seems like there should be some sort of general theory of the way that roots of families of polynomials "move around" in the plane. (And if there isn't, why not?) Does the situation I've described ring a bell for anybody?