  Previous analyses of Laguerre's method have provided results on the
convergence and properties of this popular method when applied to the
polynomials $p_n(z)=z^n-1$, $n\in\mathbb{N}$. While these analyses appear to
provide a fairly complete picture, careful study of the results reveals that
more can be said. We provide additional analytical, computational, and
graphical results, details, and insights. We raise and summarize questions that
still need to be answered.
