Conversational Speech Synthesis (CSS) is a key task in the user-agent interaction area, aiming to generate more expressive and empathetic speech for users. However, it is well-known that "listening" and "eye contact" play crucial roles in conveying emotions during real-world interpersonal communication. Existing CSS research is limited to perceiving only text and speech within the dialogue context, which restricts its effectiveness. Moreover, speech-only responses further constrain the interactive experience. To address these limitations, we introduce a Conversational Speech-Visual Synthesis (CSVS) task as an extension of traditional CSS. By leveraging multimodal dialogue context, it provides users with coherent audiovisual responses. To this end, we develop a CSVS system named UniTalker, which is a unified model that seamlessly integrates multimodal perception and multimodal rendering capabilities. Specifically, it leverages a large-scale language model to comprehensively understand multimodal cues in the dialogue context, including speaker, text, speech, and the talking-face animations. After that, it employs multi-task sequence prediction to first infer the target utterance's emotion and then generate empathetic speech and natural talking-face animations. To ensure that the generated speech-visual content remains consistent in terms of emotion, content, and duration, we introduce three key optimizations: 1) Designing a specialized neural landmark codec to tokenize and reconstruct facial expression sequences. 2) Proposing a bimodal speech-visual hard alignment decoding strategy. 3) Applying emotion-guided rendering during the generation stage. Comprehensive objective and subjective experiments demonstrate that our model synthesizes more empathetic speech and provides users with more natural and emotionally consistent talking-face animations.