Let $a,b,c\in \mathbb{R}$. We want to find $x,y,z,w$ in the following equations(according to $a,b,c$):
$$\begin{align} x^{2}+y^{2}+z^{2}+w^{2}&=1 \tag{1}\\ x^{2}+y^{2}-z^{2}-w^{2}&=a \tag{2}\\ xw+yz&=2b \tag{3}\\ yw-xz&=2c.\tag{4} \end{align}$$
(In my other post, that problem had a condition. But in this post, there is not.)
Using complex numbers, let $p:=x+iy$, and $q:=w+iz$.
From the first two equations, you can easily derive the values of $|p|$ and $|q|$ (as $\sqrt{(1\pm a)/2}$). The last two equations can be combined as $$pq^*=2(b+ic).$$
Then you can obtain the phase shift between $p$ and $q$ (which is the argument of $b+ic$) and their moduli, but the absolute phases remain undetermined.
In addition, the compatibility condition $$\frac{\sqrt{1-a^2}}2=2\sqrt{b^2+c^2}$$ must be fulfilled.