Forecasting the consumer price index (CPI) is essential for inflation monitoring and policy design, yet existing studies often rely on limited economic variables and single-country settings. This study develops and compares machine learning, deep learning, and hybrid models for cross-country CPI forecasting using a country-year panel dataset covering 1995–2020. The final sample includes 545 observations with predictors grouped into five domains: economic, political, social, environmental, and educational. To evaluate the incremental contribution of multidimensional information, variables were added sequentially from an economic baseline to broader model specifications. Across five machine learning models and five deep learning architectures, deep learning models consistently outperformed machine learning models, with temporal fusion transformer and long short-term memory showing the strongest performance. A hybrid model integrating LSTM-based sequence learning with boosting algorithms and wavelet decomposition achieved the best overall results. The highest predictive accuracy was observed when political and social variables were added to the economic baseline, whereas environmental and educational variables provided smaller but stable gains in robustness. These findings show that CPI forecasting can be improved by combining multidimensional predictors with hybrid deep learning, and they highlight the importance of institutional and labor-market conditions in explaining cross-country inflation dynamics.