Pricing high-dimensional basket options poses significant challenges, especially when dealing with nonlinear payoffs. Previous research has demonstrated the effectiveness of neural networks in pricing Bermudan basket call options, particularly under the assumption that the underlying assets follow geometric Brownian motion (GBM). Building on these, the contribution of this study is twofold. First, we extended the scope of options to include both call and put options, as well as payoffs based on the maximum and minimum of the underlying assets. Second, and more importantly, we addressed the practical relevance of market conditions by modeling the underlying assets using a high-dimensional Heston stochastic volatility model with a full correlation structure. Using a least-squares Monte Carlo approach, we approximated the continuation value of the options across a large number of underlying assets using a shallow neural network. We demonstrated that our model yields accurate results in low-dimensional Heston settings and high-dimensional GBM settings, aligning with existing literature and providing confidence in its validity. While high-dimensional pricing has been explored under GBM, our contribution lies in extending this capability to the Heston model, for which we presented numerical experiments involving up to 50 assets, a setting that, to the best of our knowledge, has not been previously studied.