Theoretical point of you. Please try to take a square root of negative frequencies and it will take you to strange place.
Some 300 years ago it was necessary to introduce a variable called $j$
Nevertheless, Laplace transform, transforms time-domain signal to $s$-domain where $$s=\sigma+j\omega$$
where as Fourier transform, to frequency domain $j\omega$
One answer is very simple: Information.
An AC signal simply can't be quantified with a single number. Sum two 100hz 1V signals and you might get anything between 0 and 2 depending on phase. Complex numbers solve this problem by 'carrying around' two pieces of information all the time.
Poles and zeros are similar - their frequency doesn't tell you ...