The general method works as \(\displaystyle z=a+bi\).
For example \(\displaystyle z=\frac{1}{i}=\frac{i}{i^2 } = \frac{i}{-1}=-i\).
Now \(\displaystyle a=0\) and \(\displaystyle b=-1\).

Yeah, that's pretty much it; makes it easy to separate the real and imaginary parts. Another benefit is it puts complex numbers in a sort of standardised form that makes it easier to compare them -- the same way we do with other radicals.

I agree with half of this. There's no mathematical reason whatsoever. But it's the standard convention, just like putting radicals in the numerator. Conventions do have meaning, such as for example driving on the right or left according to your country's laws. There's no reason for it, but you get in big trouble if you don't do what everyone else does.

I don't know about mathematical reasons, but there are definitely practical reasons. We use the poles of transfer functions to determine stability of systems. If the real part of a pole is negative, it is stable in that mode. The more negative it is, the more stable it is.
If you find a pole at $\displaystyle \frac{3\pm 5i}{-1\pm i}$, it is far from obvious whether the real part is positive or negative. If you write it as $1\pm 4i$, it is much more evident that you have an instability.