I can recommend the book Computational Complexity Theory,
edited by Steven Rudich and Avi Wigderson, which is based on a graduate summer school by the IAS/Park city mathematics institute, with lectures by various experts in the field.
The purpose of this book is to provide the basics, some history and a glimpse into the research in some areas of computational complexity theory, aimed at mathematics students.
As such, this book is not a replacement for an introductory book such as Sisper's, but can be read independently to get a feeling what some sub-fields of complexity theory look like, and can help to guide a future focus.
For completeness, I'll list a summary of the table of contents as well (see this Google books preview for the full table):
COMPLEXITY THEORY: FROM GÖDEL TO FEYNMAN
Steven Rudich, Complexity Theory: from Gödel to Feynmann
Avi Wigderson, Average case Complexity
Sanjeev Arora, Exploring Complexity through Reductions
Ran Raz, Quantum Computation
Ran Raz, Circuit and Communication Complexity
Paul Beame, Proof Complexity
RANDOMNESS IN COMPUTATION
Oded Goldreich, Pseudorandomness - Part I
Luca Trevisan, Pseudorandomness - Part II
Sadil Vadhan, Probabilistic Proof Systems - Part I
Madhu Sudan, Probablistically Checkable Proofs