Why complex numbers are represents as a+ib and can't be as (a,b)?
Yes, in signal processing, complex numbers are usually visualized on the complex plane, as you have said. 

The reason is that if you put them on a plane, then you are able to measure two important quantities: 

1) Magnitude, which is $\sqrt{x^2 + y^2}$

2) Phase angle between your point and the origin, given by $\tan^{-1} \frac{y}{x}$. 

If you simply left them as a point, ($x$,$y$), you would not be able to concretize and have a frame work for those quantities. 

You may ask, why are those quantities, in turn, important? In signal processing, we are of course dealing with signals, and physically, we are dealing with 'real' signals. However, though a nice trick, an constant oscillation of a quantity in 'real' life, (like a cosine wave), is equivalent to two phasors, rotating around in opposite directions on the complex plane, and adding up together. With this framework, we can see that the phase angles 'cancel' each other out, and that the magnitudes of their resultant give us the magnitude of our 'real' signal. 

In fact this is what one of euler's formulas captures. That is:

$$
\cos(2\pi ft) = \frac{e^{j2\pi ft} + e^{-j2\pi ft}}{2}
$$

You can see here how we can easily relate a 'real' world concept, like an oscillating cosine wave, with the 'complex' world of phasors, as they exist and rotate around in the complex plane. 

This is one of the corner stones of DSP.