What about a set of expressions in the quartic's coefficients that discriminate between all cases?
There are 9 cases:
4 distinct real roots.
3 distinct real roots with one of them being a double root.
2 distinct double roots both real.
Triple root and and a distinct fourth root.
2 distinct real roots and two complex roots.
double real root and 2 complex roots.
2 double roots both complex.
four distinct complex roots.
Such sets are known for quadratic and cubic polynomials:
Quadratic ($ ax^2+bx+c $):
Positive for two distinct real roots, zero for double root, and negative for complex conjugate roots.
Cubic ($ ax^3+bx^2+cx+d $):
$\Delta_1=2b^3-9abc+27a^2d$ and $\Delta_2=\Delta_1^2-4(b^2-3ac)^3$.
$ \Delta_2>0 $ gives one real root and two complex roots.
$ \Delta_2<0$ gives three distinct real roots.
$ \Delta_2=0 $ but $ \Delta_1\neq 0$ gives a double root plus one different root.
$ \Delta_1=\Delta_2=0$ gives a triple root.