Channel extrapolation has attracted wide attention due to its potential to acquire channel state information (CSI) with high accuracy and minimal overhead. This is becoming increasingly crucial as the sixth-generation (6G) mobile networks aim to support complex scenarios, for example, high-mobility communications utilizing ultra-massive multiple-input multiple-output (MIMO) technologies and broad spectrum bands, necessitating multi-domain channel extrapolation. Current research predominantly addresses channel extrapolation within a single domain, lacking a comprehensive approach to multi-domain channel extrapolation. To bridge the gap, we propose the concept of multi-domain channel extrapolation, detailing the essential performance requirements for 6G networks. These include precise channel extrapolation, adaptability to varying scenarios, and manageable computational complexity during both training and inference stages. In light of these requirements, we elaborate the potential and challenges of incorporating generative artificial intelligence (GAI)-based models for effective multi-domain channel extrapolation. Given the ability of the Transformer to capture long-range dependencies and hidden patterns, we propose a novel Transformer encoder-like model by eliminating the positional encoding module and replacing the original multi-head attention with a multilayer perceptron (MLP) for multi-domain channel extrapolation. Simulation results indicate that this model surpasses existing baseline models in terms of extrapolation accuracy and inference speed. Ablation studies further demonstrate the effectiveness of the module design of the proposed design. Finally, we pose several open questions for the development of practical GAI-based multi-domain channel extrapolation models, including the issues of explainability, generalization, and dataset collection.