The query suggestion or auto-completion mechanisms help users to type less while interacting with a search engine. A basic approach that ranks suggestions according to their frequency in the query logs is suboptimal. Firstly, many candidate queries with the same prefix can be removed as redundant. Secondly, the suggestions can also be personalised based on the user's context. These two directions to improve the aforementioned mechanisms' quality can be in opposition: while the latter aims to promote suggestions that address search intents that a user is likely to have, the former aims to diversify the suggestions to cover as many intents as possible. We introduce a contextualisation framework that utilises a short-term context using the user's behaviour within the current search session, such as the previous query, the documents examined, and the candidate query suggestions that the user has discarded. This short-term context is used to contextualise and diversify the ranking of query suggestions, by modelling the user's information need as a mixture of intent-specific user models. The evaluation is performed offline on a set of approximately 1.0M test user sessions. Our results suggest that the proposed approach significantly improves query suggestions compared to the baseline approach.