Googles PageRank has been called the billion dollar eigenvector; the New York Times calls it the key invention of Googles founders Larry Page and Sergey Brin. In The Book of mathematical algorithms, PageRank and the FFT decoder for the 4-by-3 telephone touchpad tone matrix must surely be in the first chapter. The Google matrix is a real nonnegative stochastic matrix, so Markov chains and Perron-Frobenius theory are a natural setting in which to analyze it. However, mathematics is rich with examples in which expanding the natural setting of a problem actually simplifies and clarifies the analysis. We discuss how basic methods of complex matrix analysis provide a better and more elementary context in which to understand PageRank. Page and Brins original paper (unpublished: not a smart step toward tenure) is available online: just Google page brin pagerank and a link to their 1998 Stanford technical report is the first one displayed.