If we have $R^{4}$ with basis $\{e_{1},e_{2},e_{3}=Je_{1},e_{4}=Je_{2}\}$, then we know that $\{e_{1},e_{2}\}$ is totally real minimal submanifold of $R^{4}$. Is there a nontrivial example of totally real minimal submanifold of $R^{4}$.
Also, the above totally real minimal immersion is an inclusion map. How we can write a basis of a submanifold in the case when the immersion $f:R^{2}\to R^{4}$ is defined as $f:(x,y)\to (x,y,x,y)$.
Yes, there are many such examples.
A totally real submanifold in this case is simply a surface $S\subset\mathbb{R}^4$ such that, for each $p\in S$, the space $J(T_pS)$ is orthogonal to $T_pS$. In other words, if $x^i$ are the coordinates dual to your basis $e_i$, then the $2$-form $\Upsilon_1 = dx^1\wedge dx^3 + dx^2\wedge dx^4$ should vanish on $S$. Suppose that we also require that the $2$-form $\Upsilon_2 = dx^1\wedge dx^4 - dx^2\wedge dx^3$ should vanish on $S$. These are equivalent to the vanishing of $$ \Upsilon_1 + i\,\Upsilon_2 = (dx^1 - i\,dx^2)\wedge (dx^3 + i\,dx^4) = dz^1\wedge dz^2 $$ where $z^1 = x^1-i x^2$ and $z^2 = x^3+ix^4$ are complex coordinates with respect to a different complex structure $J'$ on $\mathbb{R}^4$ than the one you defined, but that is also orthogonal with respect to the underlying metric on $\mathbb{R}^4$.
Any $J'$-complex curve (and there are many of them, just take $z^2 = h(z^1)$ where $h$ is holomorphic, for example), is then an example of a surface on which $\Upsilon$ vanishes. All $J'$-complex curves are minimal (since $J'$ is orthogonal), so this furnishes many examples of nontrivial totally real minimal surfaces in $\mathbb{R}^4$.
