Large Language Models (LLMs) have become integral to software engineering workflows, yet their effectiveness degrades significantly in multi-turn conversations. Recent studies demonstrate an average 39% performance drop when instructions are delivered across multiple turns, with models making premature assumptions and failing to course correct (Laban et al., 2025). This degradation is particularly problematic in exploratory programming tasks where developers need to investigate alternative approaches without committing to a single path. Current solutions force users into a false dichotomy: continue in a context-polluted conversation where the LLM becomes increasingly confused, or start fresh and lose all accumulated context. We present ContextBranch, a conversation management system that applies version control semantics to LLM interactions. ContextBranch provides four core primitives--checkpoint, branch, switch, and inject--enabling users to capture conversation state, explore alternatives in isolation, and selectively merge insights. We evaluate ContextBranch through a controlled experiment with 30 software engineering scenarios featuring intentionally polluting explorations. Branched conversations achieved higher response quality compared to linear conversations, with large improvements in focus and context awareness. Benefits were concentrated in complex scenarios involving conceptually distant explorations. Branching reduced context size by 58.1% (31.0 to 13.0 messages), eliminating irrelevant exploratory content. Our work establishes conversation branching as a fundamental primitive for AI-assisted exploratory work, demonstrating that isolation prevents context pollution when exploring alternatives.