A Pure Imaginary Bottom

When the bottom is only imaginary, like 2i, its real part is 0, so its conjugate is -2i. That works: $\left(2i\right)\left(-2i\right) = -4{i}^{2}$, which is 4, a real number.

Multiplying by i works too, with smaller numbers: $\left(2i\right)\left(i\right) = 2{i}^{2}$, which is -2. Any nonzero real multiple of i makes the bottom real.

Take $\frac{8-6i}{2i}$. Times i on top and bottom: the top is $8i-6{i}^{2}$, which is $6+8i$, and the bottom is -2. Split the fraction: 6 over -2 is -3, and 8i over -2 is -4i, so the answer is $-3-4i$. The bottom is negative, so every sign of the top flips.