When you are applying Newton's method to an equation $f(z)=0$ or to a system of equations $f_i({\bf x})=0$ $\ (1\leq i\leq n)$ defined in some region $\Omega\subset{\mathbb R}^n$ you have to be aware the that the geometric situation defined by these equations might be complicated to begin with. Now Newton's method $z_n\to z_{n+1}$ defines a discrete dynamic system on $\Omega$ with various basins of attraction of individual solutions, basins of attraction of periodic orbits and worse things. In addition these basins are intertwined in a fractal way, see, e.g. Peitgen/Saupe: The science of fractal images, pp. 207ff., for a picture of the basins for Newton's method applied to the equation $z^3-1=0$.
These things are usually not dealt with in numerical practice. One just starts with an approximation to a solution that was arrived at by heuristic or other means, and after a few runs of the algorithm it becomes obvious whether one has convergence or not.