Image search is an essential and user-friendly method to explore vast galleries of digital images. However, existing image search methods heavily rely on proximity measurements like tag matching or image similarity, requiring precise user inputs for satisfactory results. To meet the growing demand for a contemporary image search engine that enables accurate comprehension of users' search intentions, we introduce an innovative user intent expansion framework. Our framework leverages visual-language models to parse and compose multi-modal user inputs to provide more accurate and satisfying results. It comprises two-stage processes: 1) a parsing stage that incorporates a language parsing module with large language models to enhance the comprehension of textual inputs, along with a visual parsing module that integrates an interactive segmentation module to swiftly identify detailed visual elements within images; and 2) a logic composition stage that combines multiple user search intents into a unified logic expression for more sophisticated operations in complex searching scenarios. Moreover, the intent expansion framework enables users to perform flexible contextualized interactions with the search results to further specify or adjust their detailed search intents iteratively. We implemented the framework into an image search system for NFT (non-fungible token) search and conducted a user study to evaluate its usability and novel properties. The results indicate that the proposed framework significantly improves users' image search experience. Particularly the parsing and contextualized interactions prove useful in allowing users to express their search intents more accurately and engage in a more enjoyable iterative search experience.