A small note on extending the argument I gave in the previous (linked) thread.

You get a free involution on $\mathbb CP^{2n+1}$ by using the fibrewise antipodal map for your bundle $$ S^2 \to \mathbb CP^{2n+1} \to \mathbb HP^n$$ so this also gives you $\mathbb CP^{2n+1}$ as the boundary of a mapping cylinder.

$\mathbb HP^{2n+1}$ I'm not sure how to deal with analogously. I suppose a place to start would be to try and find a somehow more natural free involution on $\mathbb CP^{2n+1}$.

A small note on extending the argument I gave in the previous (linked) thread.

You get a free involution on $\mathbb CP^{2n+1}$ by using the fibrewise antipodal map for your bundle $$ S^2 \to \mathbb CP^{2n+1} \to \mathbb HP^n$$ so this also gives you $\mathbb CP^{2n+1}$ as the boundary of a mapping cylinder.

$\mathbb HP^{2n+1}$ I'm not sure how to deal with analogously. I suppose a place to start would be to try and find a somehow more natural free involution on $\mathbb CP^{2n+1}$.

Googling around it's not clear to me whether or not it's known if $\mathbb HP^{2n+1}$ admits a free involution.