I'd highly recommend Tristan Needham's Visual Complex Analysis to anyone interested in the subject, for exactly this reason . . . and, subsequently, Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables as proof that, nevertheless, "algebraic" need not imply "boring" or "computational". As a silly example, I'll never forget Cartan's definition of "2π" as the unique positive real number such that the kernel of the homomorphism "t -> e^it" from the additive group of reals to the multiplicative group of unit-length complexes is the set of integer multiples of 2π.