Here are three domain plots of a complex polynomial of degree $5$. The left picture is very zoomed out, and the right picture is more zoomed into the zeroes. (Pictures are taken from Elias Wegert's book "Visual Complex Functions".) The color indicates the argument of the function; the modulus is not featured.
Say we focus our attention on one color, like yellow. Then we can see that the yellow lines coming in from infinity seem to "end" at the zeroes of the polynomial. My confusion is: I cannot justify why this should be the case in general. Why is it true that every yellow line (that is, a line of constant argument) coming in from infinity should terminate at a zero of the polynomial?
It makes sense to me that around a zero of order $n$, the polynomial should look like $z^n$ -- that is, there should be $n$ yellow lines emanating from that point. What is not clear to me, though, is why every yellow line coming in from infinity should terminate at a zero in particular. Why can't it terminate at any other point? Any suggestions / hints would be greatly appreciated. Thanks!