In mathematics, let’s assume the original basis is $(1,0)$ and $(0,1)$. We can write $(1,1)$ as a linear combination of the original basis: $(1,1)=x(1,0)+y(0,1)$ where $x=1$, $y=1$. Also we can write $(1,-1)$ as a linear combination of the original basis: $(1,-1)=x(1,0)+y(0,1)$ where $x=1$, $y=-1$.
Now let’s assume $(1,1)$ and $(1,-1)$ is the new basis. We can write any vector let’s say $(2,1)$ as a linear combination of the new basis: $(2,1)=x(1,1)+y(1,-1)$ where $x=\frac32$, $y=\frac12$; also acknowledging that $(2,1)$ can also be written in the original basis as $(2,1)=x(1,0)+y(0,1)$ where $x=2$, $y=1$.
In effect, every vector can be written as a linear combination of any 2 vectors.
Extending the same discussion to the complex plane, and treating the 90° line as the imaginary axis/dimension,
$2+3j$ can be written as the linear combination of 2 basis vectors $1=1+0j$ and $j=0+1j$ as $$ 2+3j = x(1+0j) + y(0+1j)\qquad\text{ where }x=2\text{ and }y=3\,. $$ We make a special note here that in the case of the complex plane, $x$ and $y$ are real, whereas the basis vectors are complex numbers.
NOW, if we extend the same discussion to Quantum Computing to describe the state of a qubit, we describe the state of 1 qubit as $|q\rangle=\alpha|0\rangle+\beta|1\rangle$ where $|0\rangle$ and $|1\rangle$ can be explained according to another accept answer on Stack exchange for a different question. The numbers in notations like $|n\rangle$ are the analogues of indices in matrix notation. That is, $|0\rangle=e_0$, $|1\rangle=e_1$, etc., where $e_n$ is the vector which has a $1$ in the $n$-th position and $0$ in the other entries. For qubits in quantum computers, the dimension is 2, so we have $|0\rangle=e_0=(1,0)$ and $|1\rangle=e_1=(0,1)$. Accordingly - I am assuming – \begin{align*} |0\rangle= e_0 &= (1,0) = (\,\text{Yes_it_is_Zero}\,,\, \text{No_it_is_not_1}\,) \quad \text{ and}\\ |1\rangle= e_1 &= (0,1) = (\,\text{No_it_is_not_Zero}\,,\, \text{Yes_it_is_1}\,) \end{align*} This also means that $|0\rangle$ and $|1\rangle$ themselves are not complex unless each of them are represented as a superposition of $|0\rangle$ and $|1\rangle$ with $\alpha$ and $\beta$.
Question 1: So, if $|0\rangle$ and $|1\rangle$ themselves are not complex, then, why do we need $x$ and $y$, i.e. $\alpha$ and $\beta$ to be complex here? This is different from the note earlier for the complex plane, where the basis was complex and the $x,y$ were real numbers. What is the need for probability amplitudes to be complex numbers? What is the intuition behind probability amplitudes $\alpha$ and $\beta$ to be complex numbers?
Question 2: What is the intuition behind probability amplitudes being squared to find probabilities of $|0\rangle$ state (with $|\alpha|^2$) or $|1\rangle$ state (with $|\beta|^2$).
Question 3: And what is the meaning of pure states versus mixed states in my current framework of understanding?