I understand that Introducing complex numbers to the amplitude allows us an extra degree of freedom. Through the rotation of the complex vector, you can encode the same magnitude (1/√2) with an infinite number of configurations around a circle. The direction of these vectors is what we refer to as a phase factor.
My Question: Why the phase dimension has to be complex, can't we achieve the same extra degree of freedom with introduction of one more real dimension ?
You could. Indeed, there are computational schemes that do just that. Remember that this is a mathematical description of physical reality. So, the need for something cannot be understood entirely in an abstract manner: it has to describe experimental results. From there, the mathematical technique is a choice/convention/historical quirke out of the set of possible things that work (or are mathematically equivalent to each other, as in this case).
I guess one of the main steers in the direction of complex numbers comes from the origin of quantum mechanics. While in quantum computation, we use a lot of discrete notation (vectors, matrices,...), much of the field originates from the continuous case (and having started to look at waves). There, it is far more natural to use complex numbers rather than adding an extra degree of freedom.
