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Let's say we have a complex differentiable function $f(x)$ that has a complex conjugate pair of roots at some point, lets say 5+i and 5-i for instance.

let $g(x)$ be a fixed-point iteration which gives $g(x)=x$ when $f(x)=0$, say, the Newton derivative, or some other fixed-point iteration like Steffensen, etc.

Is it possible to converge to one of the complex roots, if the iteration is started from a point which is equal to the midpoint of the imaginary parts?

Another way to say this is, would the influences of the pair of roots on the trajectories of the iteration cancel each other such that the iteration would not be influenced by the pair and hence not converge to either of them?

of course it would converge if started to the left or right of the midpoint, but it seems that exactly on it, they might cancel

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    Yep... I tried the iteration on the functions mentioned here, http://math.stackexchange.com/questions/2161639/complex-valued-function-that-is-real-for-real-arguments-but-also-has-complex-roo/2161644#2161644 and indeed, if its positioned directly on the midpoint it will not converge to either of the imaginary roots and the trajectories tend to behave rather wildly and diverges if there isnt a root on a line perpendicular to the alignment of the conjugate pair – 2017-02-26
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    @Moo - that should be "no real numerical routine". If the routine itself involves complex numbers, then it has no difficulty picking them up from a real starting point. – 2017-02-26

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