I have something like this $$z+{\frac{1}{i}}$$ Which I expand: $$z+{\frac{1}{i}} = x+yi+{\frac{1}{i}} = x+i(y+?)$$ I'm not sure what should be in place of $?$.
Assuming $i\times x = {\frac{1}{i}}$, then $x = {\frac{1}{i}} \div i = {\frac{1}{i}} \times {\frac{1}{i}} = -1$. But definitely, $i\times (-1) \neq {\frac{1}{i}}$.
We have
$$z+{\frac{1}{i}} = x+yi+{\frac{1\cdot i}{i\cdot i}} = x+yi+{\frac{i}{-1}}= x+iy-i=x+(y-1)i$$
and more in general for any $z\in \mathbb{C}\, z\neq 0$
$$\frac1{z}=\frac1{z}\frac{\bar z}{\bar z}=\frac{\bar z}{z \bar z}=\frac{\bar z}{\mid z \mid^2}$$