The parameter alpha in Quantile Regression, similar to Lasso in linear regression, controls the strength of the L1 regularization, which adds a penalty to the model's complexity like this site. Regularization helps avoid overfitting by reducing the influence of less significant features. In higher-dimensional problems, alpha can shrink less important dimensions, leading to a more interpretable model with fewer, more predictive features. So, your intuition about it affecting multiple dimensions is on track.