Consider $\mathbb{R}^2$ in $\mathbb{C}^2$ (natural embedding). Need show if $f \in O(\mathbb{C}^2 - \mathbb{R}^2)$ then it extends to a holomorphic function of $\mathbb{C}^2$.
The hint that has been given: Change coordinates before using Hartogs.
My attempt: Shift $\mathbb{R}^2$ by translating it by $(i,i)$ call it $A$. Then in the absolute space $A$ would be forced to have $|z_1| \geq 1$ and $|z_2| \geq 1$. After this, I am not able to figure out an appropriate Hartogs figure. This would have worked if we were to remove at $\mathbb{R} \times \{\lambda_o\}$ from $\mathbb{C}^2$ instead; translate it by $(i,-\lambda_o)$ and a Hartogs figure can be found.
I also tried to use Analytic discs. The line $\{(z_1+\lambda, \lambda): \lambda\in\mathbb{C}\}$ doesn't intersect $\mathbb{R}^2$ when $Im(z_1)\neq 0$. Slowly moving $z_1$ over $\mathbb{C}$ doesn't intersect $\mathbb{R}^2$ nicely. As $Im(z_1)=0$ the line intersects $\mathbb{R}^2$ in a non compact set. A more general equation of line yielded a similar observation.