Yes, it is just a short-cut. The point is that the complexified map
$$\tag{1} \begin{pmatrix} \phi \\ \phi^{*} \end{pmatrix} ~=~ \begin{pmatrix} 1 & i\\ 1 &-i \end{pmatrix} \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} $$
is a bijective map :$\mathbb{C}^2 \to \mathbb{C}^2 $.
Originally $\phi_1,\phi_2 \in \mathbb{R}$ are of course two real fields. But we can complexify them, vary the action wrt. the two independent complex variables $\phi_1,\phi_2 \in \mathbb{C}$, if we at the end of the calculation impose the two real conditions $$\tag{2} {\rm Im}(\phi_1)~=~0~=~{\rm Im}(\phi_2). $$
Or equivalently, we can treat $\phi$ and $\phi^{*}$ as two independent complex variables, vary the action wrt. the two independent complex variables $\phi,\phi^{*} \in \mathbb{C}$, if we at the end of the calculation impose the complex condition
$$\tag{3} \phi^{*} ~=~ \text{complex conjugate of }\phi. $$
The Euler-Lagrange equations that we derive via the two methods (1) and (2) will be just linear combinations of each other given by the constant matrix from eq. (1).
