How to handle points in extended finite field
As you note, the elements of $\mathbf F_{p^2}$ can be represented as $a+bi$  with $a,b \in \mathbf F_p$ and $i^2 = -1$; however note that this only works if $p\equiv 3 \pmod 4$.

Then you can add and multiply two elements of $\mathbf F_{p^2}$ in basically the same way you add and multiply complex numbers. Namely, $(a+bi)+(c+di) = (a+c)+(b+d)i$ and $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$.

And the formula for point addition is the same, but you need to keep in mind that all your "numbers" are elements of $\mathbf F_{p^2}$ and must be handled as above.