LLM-driven Anomaly Detection (AD) helps enhance the understanding and explanatory abilities of anomalous behaviors in Time Series (TS). Existing methods face challenges of inadequate reasoning ability, deficient multi-turn dialogue capability, and narrow generalization. To this end, we 1) propose a multi-agent-based TS Evolution algorithm named TSEvol. On top of it, we 2) introduce the AD reasoning and multi-turn dialogue Dataset TSEData-20K and contribute the Chatbot family for AD, including ChatAD-Llama3-8B, Qwen2.5-7B, and Mistral-7B. Furthermore, 3) we propose the TS Kahneman-Tversky Optimization (TKTO) to enhance ChatAD's cross-task generalization capability. Lastly, 4) we propose a LLM-driven Learning-based AD Benchmark LLADBench to evaluate the performance of ChatAD and nine baselines across seven datasets and tasks. Our three ChatAD models achieve substantial gains, up to 34.50% in accuracy, 34.71% in F1, and a 37.42% reduction in false positives. Besides, via KTKO, our optimized ChatAD achieves competitive performance in reasoning and cross-task generalization on classification, forecasting, and imputation.