Basic Properties of Analytic Functions
Cauchy-Riemann equations
Elementary functions, branches and branch cuts
Elementary Topology, connected sets
Analytic functions and Harmonic functions
Inverse function Theorem

Closed and exact I-forms

Harmonic Conjugates

Cauchy's Theorem
Contour integrals, Path Independence
Cauchy's Theorem
Cauchy's integral formula (for the function and derivatives)
Liouville's Theorem, Fundamental Theorem of Algebra
Morera's Theorem
Goursat's Theorem
Mean Value Theorems
Local and Global Maximum Moduli Theorems

Analytic Series
Uniform Convergence: Integration and differentiation of series
Power series
Analytic continuation, Monodromy Theorem
poles, essential singularities
Laurent series
Zeros of analytic functions
Casorati-Weierstrass Theorem

Calculus of Residues and Integrals
Residue Theorem
Partial Fraction Theorem
Definite integrals via residues

Conformal Mappings
Mapping properties of elementary functions
Fractional linear transformations: inverse points, geometrical properties

Argument Principle
Rouche's Theorem
Hurwitz's Theorem
Schwartz Lemma

References:

Basic Complex Analysis, J. Marsden and M. Hoffman, W. H. Freeman and Co.
Complex Analysis, S. Lange.
Complex Analysis, T.W. Gamelin, Springer
Complex Analysis, L.V. Ahlfors