The terms that users pass to search engines are often ambiguous, by referring to homonyms. The search results in these cases are a mixture of links to documents that refer to different meanings of the search terms. Current search engines provide suggested query completion terms as a dropdown list. However, such lists are not well organized, mixing completions for different meanings. In addition, the suggested search phrases are not discriminating enough. We propose an approach to suggesting well organized and visually separated search completions based on ontologies. In addition, our approach supports the use of negative terms to disambiguate the suggested completions in the list. We present an algorithm to generate the suggested search completion terms. We have developed musician and basketball player ontologies and an Ontology-Supported Web Search (OSWS) System for "famous people" that generates the suggested search term completions, based on these ontologies.