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.