Tristan Needham has written a wonderful synthesis of geometry, complex analysis and vector fields. Before I read this book I had "studied" complex analysis, but had never truly understood it. Now it all makes sense !
The scope of the book is very broad. It covers 2D and 3D geometry, Mobius transforms, non-Euclidean geometry, analytic functions, complex differentiation and integration, winding numbers, vector fields and harmonic functions. But it is the approach that makes this text so unusual and so accessible.
Needham believes that geometric arguments reveal underlying connections which algebraic proofs diguise. In his own words: "while it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by bringing you nearer to the Truth". Needham gloriously justifies his assertion in this text. Geometric proofs are used wherever possible, with the final conclusions translated back into algebraic terms. A variety of effective techniques are introduced for visualising the effect of Mobius transforms, analytic functions, complex differentiation etc.
One small word of warning - as Needham says himself in the Introduction, the arguments in this book are not formally rigorous. He bypasses the usual scaffolding of convergence and limits, and treats continuity as an intuitive concept. He uses phrases such as "the effect on an infinitesimal vector" which would cause a sharp intake of breath from a purist. This is not a problem, as long as you are happy to take it on trust that a formal framework can be provided if required. However, if you are studying for a conventional complex analysis exam, then you will need to fill in the formal structure from a more "standard" text once you know the landscape.
Definitely one of the best maths textbooks that I have ever read.