Complex Question Answering over Knowledge Graph (C-KGQA) seeks to solve complex questions using knowledge graphs. Currently, KGQA systems achieve great success in answering simple questions, while complex questions still present challenging issues. As a result, an increasing number of novel methods have been proposed to remedy this challenge. In this survey, we proposed two mainstream categories of methods for C-KGQA, which are divided according to their use for knowledge graph representation and construction, namely, graph metric (GM)-Based Methods and graph neural network (GNN)-based methods. Additionally, we also acknowledge the influence of ChatGPT, which has prompted further research into utilizing knowledge graphs as a knowledge source to assist in answering complex questions. We also introduced methods based on pre-trained models and knowledge graph joint reasoning. Furthermore, we have compiled research achievements from the past three years to make it easier for researchers with similar interests to obtain state-of-the-art research. Finally, we discussed the resources and evaluation methods for tackling C-KGQA tasks and summarized several research prospects in this field.