Methods are presented for seeking all roots, real and complex, to single, nonlinear equations using global continuation with fixed-point and Newton homotopies. The methods are facilitated by unique properties of the homotopies when applied to a single nonlinear equation starting from a single real value. In some cases, the homotopy path must be tracked through the complex domain to obtain all roots, even in the absence of complex roots. For this purpose, a complex predictor-corrector method is developed, which tracks both real- and complex-valued paths and locates bifurcation points where real-valued paths become complex-valued paths. The methods are applied to 16 chemical engineering problems. For all problems, all real roots are obtained and all complex roots are obtained when they exist.