We begin with $\mathbb{H}^2$ and we let $[q_1,q_2]$ denote the class $\{ (q q_1, q q_2), q \in \mathbb{H}, q \neq 0\}$. Under stereographic projection, there is a biejction between this set and $S^4$. We thus a have a quaternionic vector bundle $\mathbb{V} \to S^4$ with fiber isomorphic to $\mathbb{H}$. Under the $\mathbb{C}$ linear isomorphism $\mathbb{H} \cong \mathbb{C}^2$ one can see the bundle as a complex bundle with fiber $\mathbb{C}^2$. What is $c_2(\mathbb{V})$?
According to wikipedia the Potryagin class of the quaternionic projective plane $\mathbb{HP}^2$ is given by $(1+v)^6 (1-v)^{-1}$ where $v$ is a generator of $H^4 (\mathbb{HP}^2,\mathbb{Z})$. If I am not mistaken, $\mathbb{V} = \mathbb{HP}^2$. Is this correct? If yes, maybe we could use the relation $(1-p_1(E)+p_2(E))=(1+c_1(E)+c_2(E))(1-c_1(E)+c_2(E))$.
Nevertheless, I am failing to obtain the required class and am not sure how to compute it.