Only knowledge of single variable calculus(upto Riemann sums and integration), knowing what partial derivatives, some knowledge of addition and multiplication of complex numbers and some geometry knowledge will be enough for reading this book without any problems. If you are an undergrad going to start complex analysis or anyone interested in this field, I would say this is a MUST READ. I won't say anything else, read it and you'll see. It's a joy to go through.
(P.S: Follow a standard text along with it to do some standard problems to pass exams or sometimes see the exact definitions and rigorous proofs of the theorems covered in the book, but only AFTER they have been read an understood from Visual Complex Analysis)