Quite simply, I love this book.  I came to it by way of Roger Penrose's  Road to Reality, which says some very flattering things about Needham, and being a fan of Penrose's book, I tried out Needham's.  If you loved Road to Reality, and learned something from it, there is a good chance you will also love Visual Complex Analysis.  Having said that, they are not the same kind of book.  Penrose has an end in mind, and the book arcs toward it, supplying you at each step with what you need to take the next.  Needham's book is a leisurely tour through  a particularly beautiful part of the landscape of math. You are there for the views, to stand lingering at the precipice watching the sun set.  If you an engineering or physics student interested in complex analysis because you need  to get on with your physics and engineering work, this book is not for you.  Complex Differentiation and the Cauchy-Riemann equations are not introduced until Chapter 4.  There is way more time spent on topics like the Riemann sphere and Mobius transformations than any reasonable textbook could afford.  There are extended discussions of individual functions on the complex plane, like the exponential function, the log function, the trigonometric functions.  Why?  Because they are beautiful. Because these are all places where there is an exploision of hidden structure in the complex plane.  Needham has a gift for revisiting elementary topics  and bringing out their beauty.  The discussion (and proof?) of Euler's Equation is inspiring.  All this is very untextbooklike.  Also untextbooklike: there are few exercises, and they are not really designed to work the reader through a minimal skill set.  Like the rest of the book, they focus on the areas where the views are good.  Readers turning to complex analysis for practical reasons are better off with Gamelin.  But if you want insight, and a fresh look at old ideas, and the occasional flash of purely geometric intuition, then this book is for you.  Or if you too believe that the complex plane is the doorway to some beautiful mathematics, then this book is especially for you.