I am just starting to get up to speed with quantum computing via the Quiskit learning path: online tutorial
Here they explain the Dirac notation and use it to describe quantum states as elements in $\mathbb{C}^2$ (square could be for the initial lesson but my question holds in n.
i.e a quantum state $| a \rangle = \begin{pmatrix}a_1 \\ a_2 \end{pmatrix}$ where $a_1, a_2 \in \mathbb{C}$
Shortly after this the concept of the basis is introduced, with the x, y and z examples given. Their orthonormality is stressed. In all three cases this 'basis' is two dimensional.
So as a concrete example $ \{| 0 \rangle , | 1 \rangle \}$ is given as an orthonormal basis for the space to describe and measure quantum states.
Clearly this pair of basis is not an orthonormal basis of $\mathbb{C}^2$ so I understand there may be additional constraints on the space of possible valid quantum states. But then I have not yet seen why it is necessary to embed this apparently two dimensional space into teh 4-dimension $\mathbb{C}^2$.
We have touched on the Bloch Sphere which is a two dimensional representation of pure states but that is derived from the two dimensional orthonormal basis rather than the other way around - but perhaps this is a more profound representation of the space than it seems at this point.
What is the reason we both need a 4-dimensional space to describe our possible quantum states and can work with a two dimensional basis?