In this paper, we study independent domination in directed graphs, which was recently introduced by Cary, Cary, and Prabhu. We provide a short, algorithmic proof that all directed acyclic graphs contain an independent dominating set. Using linear algebraic tools, we prove that any strongly connected graph with even period has at least two independent dominating sets, generalizing several of the results of Cary, Cary, and Prabhu. We prove that determining the period of the graph is not sufficient to determine the existence of an independent dominating set by constructing a few examples of infinite families of graphs. We show that the direct analogue of Vizing's Conjecture does not hold for independent domination number in directed graphs by providing two infinite families of graphs. We initialize the study of time complexity for independent domination in directed graphs, proving that the existence of an independent dominating set in directed acyclic graphs and strongly connected graphs with even period are in the time complexity class $P$. We also provide an algorithm for determining existence of an independent dominating set for digraphs with period greater than $1$.