What numerical analysis situations become more/less stable, have faster/slower convergence, or are otherwise quite different when dealing with functions of complex variable instead of functions of a real variable?
Complex numerical differentiation is stable, unlike real numerical differentiation.
See pages 32-33 of "Applied and Computational Complex Analysis" vol 3, Peter Henrici,
"The Complex-Step Derivative Approximation", JOAQUIM R. R. A. MARTINS, PETER STURDZA and JUAN J. ALONSO,
and this Wikipedia article on complex variable methods for numerical differentiation.