Conversational image search, a revolutionary search mode, is able to interactively induce the user response to clarify their intents step by step. Several efforts have been dedicated to the conversation part, namely automatically asking the right question at the right time for user preference elicitation, while few studies focus on the image search part given the well-prepared conversational query. In this paper, we work towards conversational image search, which is much difficult compared to the traditional image search task, due to the following challenges: 1) understanding complex user intents from a multimodal conversational query; 2) utilizing multiform knowledge associated images from a memory network; and 3) enhancing the image representation with distilled knowledge. To address these problems, in this paper, we present a novel contextuaL imAge seaRch sCHeme (LARCH for short), consisting of three components. In the first component, we design a multimodal hierarchical graph-based neural network, which learns the conversational query embedding for better user intent understanding. As to the second one, we devise a multi-form knowledge embedding memory network to unify heterogeneous knowledge structures into a homogeneous base that greatly facilitates relevant knowledge retrieval. In the third component, we learn the knowledge-enhanced image representation via a novel gated neural network, which selects the useful knowledge from retrieved relevant one. Extensive experiments have shown that our LARCH yields significant performance over an extended benchmark dataset. As a side contribution, we have released the data, codes, and parameter settings to facilitate other researchers in the conversational image search community.