This paper proposes CONVAUG, an LLM-based data augmentation framework for generalizing conversational dense retrieval. The framework generates multi-level augmented conversations to capture the diverse nature of conversational contexts. Inspired by human cognition, it designs a cognition-aware prompting process to mitigate false positives, false negatives, and hallucinations. It also develops a difficulty-adaptive sample filter to select challenging samples for complex conversations, thereby expanding the model's learning space. A contrastive learning objective is then used to train a more robust conversational context encoder. Extensive experiments on four public datasets under both normal and zero-shot settings demonstrate the effectiveness, generalizability, and applicability of CONVAUG. The code is available at https://github.com/haon-chen/ConvAug. The contributions include: (1) an LLM-based multi-level data augmentation framework for conversational search, (2) a cognition-aware prompting process to prevent false positives/negatives and mitigate hallucination issues, and (3) a difficulty-adaptive sample filter to select challenging samples for complex conversations. The framework improves the performance of various conversational dense retrievers across different levels of conversational complexity.This paper proposes CONVAUG, an LLM-based data augmentation framework for generalizing conversational dense retrieval. The framework generates multi-level augmented conversations to capture the diverse nature of conversational contexts. Inspired by human cognition, it designs a cognition-aware prompting process to mitigate false positives, false negatives, and hallucinations. It also develops a difficulty-adaptive sample filter to select challenging samples for complex conversations, thereby expanding the model's learning space. A contrastive learning objective is then used to train a more robust conversational context encoder. Extensive experiments on four public datasets under both normal and zero-shot settings demonstrate the effectiveness, generalizability, and applicability of CONVAUG. The code is available at https://github.com/haon-chen/ConvAug. The contributions include: (1) an LLM-based multi-level data augmentation framework for conversational search, (2) a cognition-aware prompting process to prevent false positives/negatives and mitigate hallucination issues, and (3) a difficulty-adaptive sample filter to select challenging samples for complex conversations. The framework improves the performance of various conversational dense retrievers across different levels of conversational complexity.