Continual Text Classification (CTC) aims to continuously classify new text data over time while minimizing catastrophic forgetting of previously acquired knowledge. However, existing methods often focus on task-specific knowledge, overlooking the importance of shared, task-agnostic knowledge. Inspired by the complementary learning systems theory, which posits that humans learn continually through the interaction of two systems -- the hippocampus, responsible for forming distinct representations of specific experiences, and the neocortex, which extracts more general and transferable representations from past experiences -- we introduce Information-Theoretic Complementary Prompts (InfoComp), a novel approach for CTC. InfoComp explicitly learns two distinct prompt spaces: P(rivate)-Prompt and S(hared)-Prompt. These respectively encode task-specific and task-invariant knowledge, enabling models to sequentially learn classification tasks without relying on data replay. To promote more informative prompt learning, InfoComp uses an information-theoretic framework that maximizes mutual information between different parameters (or encoded representations). Within this framework, we design two novel loss functions: (1) to strengthen the accumulation of task-specific knowledge in P-Prompt, effectively mitigating catastrophic forgetting, and (2) to enhance the retention of task-invariant knowledge in S-Prompt, improving forward knowledge transfer. Extensive experiments on diverse CTC benchmarks show that our approach outperforms previous state-of-the-art methods.