Suppose we have an ordinary differential equation:
$\dfrac{dx}{dt} = f(x)$
To determine the value of the fixed/critical/equilibrium point(s), we set $\dfrac{dx}{dt} = 0$, that is, we calculate the values of $x$ which satisfy the equation $f(x) = 0$.
Suppose a fixed point exists called $x_*$.
To determine the nature of this fixed point, we calculate $\dfrac{dx}{dt}$ to the left and to the right of it. If $\dfrac{dx}{dt}>0$ to the left of $x_*$ and $\dfrac{dx}{dt}<0$ to the right of $x_*$ we say that it is stable. Otherwise, it is unstable.
The analysis above was limited to real fixed points, that is $x_*$ $\in \mathbb{R}$.
How may we generalise and extend the above analysis and determine the nature of complex fixed points in one dimension?
So, suppose we have: $\dfrac{dx}{dt} = 1 + x^2$. The fixed point is complex and I cannot explain its nature.
You have first to decide what is your phase space:
If it is $\mathbb R$, then $x'=1+x^2$ has no constant solutions (what you call fixed points).
If it is $\mathbb C$, then you need to solve $z'=1+z^2$ (this is better) or to transform it in $x'=1+x^2-y^2, \ y'=2xy$ and proceed from there.