The first version (2006) of the book was announced in EMS Newsletter June 2006, No. 60, p. 8, and praised there by Roger Penrose and then voted one of Choice magazine’s Outstanding Academic Titles for 2007. The current reprinted edition 2008 includes some corrections and a CD updated for the newer Version 6 of Mathematica®. The book presents complex numbers and complex analysis with many applications in a state-of-the-art computational environment. It offers enrichment of the standard complex analysis course: Newton-Raphson iterations and complex fractals, the Mandelbrot set, complex chaos and bifurcations, Fourier and Laplace transforms, fluid dynamics, discrete Fourier and Laplace transforms, Schwarz-Christoffel mapping, mathematical art and tiling of the hyperbolic planes and physics in three and four dimensions (e.g. Minkowski space and twistors). Each chapter uses Mathematica® as a powerful and flexible tool to develop the reader’s geometric intuition and enthusiasm with perfect illustrations and checks of classical calculations (equations, residue, contour integrals, summation of series, mappings and transforms and their inverses). The sophisticated Mathematica® codes - included both in the text and on the CD - enable the user to run computer experiments. The book is far more than a standard course of complex analysis or a guide to Mathematica® tools.