How to find a starting point for root finding in complex plane? The function $f(x) = x^2 +1$ has zeros in the complex plane at $x = +i$ and $x = -i$. Is there a real starting point for complex Newton's method such that the iterates converge to either of these zeros? Is there a complex starting point too?
I know that Newton's method for finding roots is $x_{n+1} = x_n - \frac {f(x)}{f'(x)}$. But I don't know how to find the proper starting point.
 A: For a polynomial with real coefficients the Newton iteration starting from real points will always stay on the real axis, as all the arithmetic operations stay in the reals.

For polynomial functions the Newton iteration is contractive towards the area of the roots. There are counter examples of oszillating iterations, but in general the iteration will converge from a majority of complex initial points. It helps to start close to the root location as far away from them the convergence is linear with contraction factor $\approx 1-\frac1{\deg(f)}$.

See also Newton fractals to illustrate the chaotic relation between starting points and the roots the iteration converges to.

If you instead try to solve the equivalent $0=g(z)=e^{iεz}f(z)$ with the Newton step
$$
z_+=z-\frac{g(z)}{g'(z)}=z-\frac{f(z)}{f'(z)+iεf(z)}
$$
with some small real $ε$, then iterations starting close to a real solution will still converge to that real solution, however real starting points far from a solution might also converge towards complex roots.
