This is an exercise in the first chapter of "Complex Abelian Varieties" by Christina Birkenhake and Herbert Lange. As the title suggests it asks for an example of a complex torus of dimension $\geq 2$ not admitting any nontrivial complex subtorus.
The following is what I have tried for dimension 2. By a change of coordinate over $\mathbb{R}$, we can assume that the lattice is $\mathbb{Z}^4$. The question is equivalent to constructing a $4\times 4$ real matrix $J$ such that $J^2=-\mathrm{Id}$ and $J$ has no nontrivial invariant subspaces that are spanned by vectors with rational coefficients. Let $J$ be such a matrix, then for any nonzero $v_1 \in \mathbb{Q}^4$, we must have $Jv_1 \neq xv_1+yv_2$ for any $v_2\in \mathbb{Q}^4$ and $x,y \in \mathbb{R}$. To achieve this, we note that the field generated by the coordinates of $xv_1+yv_2$ has at most transcendental degree 2 over $\mathbb{Q}$, so if we can choose $J$ with many independent transcental entries such that $Jv$ have high transcendental degree over $\mathbb{Q}$ for all $v\in \mathbb{Q}$, then the problem is solved. But it is at this place I got stuck. Intuitively, I feel this is doable. But I am having trouble to write down such a matrix.
Of course, there are probably other approaches to the problem (the Jacobian of a curve in general position of genus $n\geq 2$, for example).
Thank you.
Edit: I should also assumed that $\det(J)=1$ so it preserves orientation. I think the solution probably look like this: we consider the space of all complex structures on $\mathbb{R}^4$. Each $2$ dimesnional subspace of $\mathbb{Q}^4$ defines a "subvariety" on this space, namely all $J$'s that preserve this subspace. These gives us countably many such "subvarieties". Intuition tells that these should not be able to cover the big space. But I have yet to make everything precise.