Why do we need a sphere to graphically represent a qubit? Bloch sphere is a common visualisation of possible qubit values, but I fail to understand why do we need a sphere to represent them. Assuming we want to represent the single qubit a|0> + b|1> theoretically we know both a and b coefficients may be complex. However, we also know that the global phase factor is insignificant (we treat two unit vectors related by a global phase as equivalent) so we may treat a or b to be a real number. Wouldn't a circle be enogh to represent a qubit then (like taking axis y away from the original Bloch sphere)?

I feel like I am missing something simple (or some basic information I should know), any suggested read or explanation would be appreciated. Thank you in advance.
If we have $a|0\rangle + b|1\rangle$, there are two quantities we need to know to specify the qubit: 


the ratio $a/b$, which is encoded by the latitude: $\left|0\right\rangle$ is the north pole and $\left|1\right\rangle$ the south,
the relative phase of $a$ and $b$, which gives the longitude. 


That is, you can treat either $a$ or $b$ as a real number, but not both. 

A circle can only specify one quantity.