Two (very recent) books on (at least selected aspects of) the history of complex (holomorphic) dynamics, in the years >1850:
Audin, Michèle: Fatou, Julia, Montel. The great prize of mathematical sciences of 1918, and beyond. Translated from the 2009 French original by the author. Lecture Notes in Mathematics, 2014. History of Mathematics Subseries. Springer, Heidelberg, 2011. viii+332 pp. ISBN: 978-3-642-17853-5 (the French original is from 2009)
Alexander, Daniel S.; Iavernaro, Felice; Rosa, Alessandro: Early days in complex dynamics. A history of complex dynamics in one variable during 1906–1942. History of Mathematics, 38. American Mathematical Society, Providence, RI; London Mathematical Society, London, 2012. xviii+454 pp. ISBN: 978-0-8218-4464-9
