Define an $\mathbb{H}$-bundle $\mathbb{H} \hookrightarrow \mathbb{H}P^2 - \{[0:0:1]\} \xrightarrow{~\pi~} \mathbb{H}P^2,$ $\pi([u:v:w]) = [u:v] \in \mathbb{H}P^1$ as follows. $\mathbb{H}P^1$ decomposes as the union of two disks $D_1 = \{[u:1] : \|u\| \leq 1\},$ $D_2 = \{[1:v] : \|v\| \leq 1\}$ glued along their common boundary via the map $[u:1] \mapsto [1:u^{-1}]$. Then take the local trivializations of our bundle to be $\psi_1: D_1 \times \mathbb{H} \longrightarrow \pi^{-1}(D_1),$ $\psi_1([u:1],w) = [u:1:w],$ $\psi_2: D_2 \times \mathbb{H} \longrightarrow \pi^{-1}(D_2),$ $\psi_2([1:v],w) = [1:v:w].$ Making the identifications $\mathbb{H} \cong \mathbb{R}^4$ and $\mathbb{H}P^1 \cong S^4$, this defines an $\mathbb{R}^4$-bundle over $S^4$. The transition function on the equatorial $S^3$ is $\psi_2^{-1} \psi_1([u:1],w) = \psi_2^{-1}([u:1:w]) = \psi_2^{-1}([1:u^{-1}:u^{-1}w]) = ([1:v],vw).$
Now from the total space $\mathbb{H}P^2 - \{[0:0:1]\}$ of our bundle, remove the open $8$-disk $D = \{[u:v:1] : \|u\|^2 + \|v\|^2 < 1\}$ centered at $[0:0:1]$. For every $[u:v] \in \mathbb{H}P^1$, this restricts the fiber over $[u:v]$ to $\{[u:v:w] : \|w\|^2 \leq \|u\|^2 + \|v\|^2\} \cong D^4.$ Therefore we have a disk bundle $D^4 \hookrightarrow \mathbb{H}P^2 - D \xrightarrow{~\pi~} S^4$ with transition function $\psi_2^{-1} \psi_1([u:1], w) = ([1:v], vw) \tag{$\ast$}$ on the equatorial $S^3$.
$S^3$-bundles over $S^4$ are classified by elements of $\pi_3(\mathrm{SO}(4)) \cong \mathbb{Z} \oplus \mathbb{Z}.$ An explicit isomorphism identifies the pair $(h,j) \in \mathbb{Z} \oplus \mathbb{Z}$ with the $S^3$-bundle over $S^4$ with transition function $f_{hj}: S^3 \longrightarrow \mathrm{SO}(4),$ $f_{hj}(u) \cdot v = u^h v u^j$ on the equatorial $S^3$, where here we consider $u \in S^3$ and $v \in \mathbb{R}^4$ as quaternions, i.e. the expression $u^h v u^j$ is understood as quaternion multiplication. From $(\ast)$, we see that the bundle $D^4 \hookrightarrow \mathbb{H}P^2 - D \xrightarrow{~\pi~} S^4$ has the transition function $f_{10}$. You can easily check that the Hopf bundle $S^3 \hookrightarrow S^7 \longrightarrow S^4$ has transition function $f_{10}$ as well, so by the above identification of $S^3$-bundles over $S^4$, its associated disk bundle must be isomorphic to $D^4 \hookrightarrow \mathbb{H}P^2 - D \xrightarrow{~\pi~} S^4.$ Therefore the total space of the disk bundle associated to the Hopf bundle over $S^4$ is $\mathbb{H}P^2$ with an open $8$-disk removed.