Personally, I think complex analysis is important. I really like Serge Lang's book "Complex Analysis." It is supposedly a graduate level text, but the first half of it covers the basics at a level suitable for upper division undergrad class. Ahlfor's book is also really really good, but a little harder to read in my opinion.
I didn't like either Royden or Rudin. I really like Folland's book "Real analysis: modern techniques and their applications." His Fourier analysis book is also pretty good IMO.
Yes, I've got a copy of Alfors. And at one time I started a course from a student of Alfors. And I have a book by Hille.
I just never could see much utility in complex variables: Complex valued functions of real variables. Sure. Complex valued measures? Sure. Fourier theory making a lot of use of complex numbers? Of course. A vector space where the field is the complex numbers? Certainly. Hermitian and unitary matrices with complex numbers? About have to like those due to the fundamental theorem of algebra, that is, roots of polynomials and, thus, complex eigenvalues. Functions of a complex variable? Never could see the utility.
I didn't like either Royden or Rudin. I really like Folland's book "Real analysis: modern techniques and their applications." His Fourier analysis book is also pretty good IMO.