Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently?
Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the underlying vector bundle and let $p_k(\xi^{\mathbb{H}})=p_k((\xi^{\mathbb{H}})_{\mathbb{R}})=(-1)^k c_{2k}((\xi^{\mathbb{H}})_{\mathbb{R}}\otimes \mathbb{C})$. But in that way we ignored the quaternionic structure and lost the key information...