  We construct an explicit map from a generic minimal $\delta(2)$-ideal
Lagrangian submanifold of $\mathbb{C}^n$ to the quaternionic projective space
$\mathbb{H}P^{n-1}$, whose image is either a point or a minimal totally complex
surface. A stronger result is obtained for $n=3$, since the above mentioned map
then provides a one-to-one correspondence between minimal $\delta(2)$-ideal
Lagrangian submanifolds of $\mathbb{C}^3$ and minimal totally complex surfaces
in $\mathbb{H}P^2$ which are moreover anti-symmetric. Finally, we also show
that there is a one-to-one correspondence between such surfaces in
$\mathbb{H}P^2$ and minimal Lagrangian surfaces in $\mathbb{C}P^2$.
