Large language models (LLMs) with integrated search tools show strong promise in open-domain question answering (QA), yet they often struggle to produce complete answer set to complex questions such as Which actor from the film Heat won at least one Academy Award?, which requires (1) distinguishing between multiple films sharing the same title and (2) reasoning across a large set of actors to gather and integrate evidence. Existing QA benchmarks rarely evaluate both challenges jointly. To address this, we introduce DeepAmbigQAGen, an automatic data generation pipeline that constructs QA tasks grounded in text corpora and linked knowledge graph, generating natural and verifiable questions that systematically embed name ambiguity and multi-step reasoning. Based on this, we build DeepAmbigQA, a dataset of 3,600 questions requiring multi-hop reasoning and half of them explicit name ambiguity resolving. Experiments reveal that, even state-of-the-art GPT-5 show incomplete answers, achieving only 0.13 exact match on ambiguous questions and 0.21 on non-ambiguous questions. These findings highlight the need for more robust QA systems aimed at information gathering and answer completeness.