Influence maximization is a well-studied problem that asks for a small set of influential users from a social network, such that by targeting them as early adopters, the expected total adoption through influence cascades over the network is maximized. However, almost all prior work focuses on cascades of a single propagating entity or purely-competitive entities. In this work, we propose the Comparative Independent Cascade (Com-IC) model that covers the full spectrum of entity interactions from competition to complementarity. In Com-IC, users' adoption decisions depend not only on edge-level information propagation, but also on a node-level automaton whose behavior is governed by a set of model parameters, enabling our model to capture not only competition, but also complementarity, to any possible degree. We study two natural optimization problems, Self Influence Maximization and Complementary Influence Maximization, in a novel setting with complementary entities. Both problems are NP-hard, and we devise efficient and effective approximation algorithms via non-trivial techniques based on reverse-reachable sets and a novel "sandwich approximation". The applicability of both techniques extends beyond our model and problems. Our experiments show that the proposed algorithms consistently outperform intuitive baselines in four real-world social networks, often by a significant margin. In addition, we learn model parameters from real user action logs.