I do not understand, why can some molecules absorb all UV light, even if they do not have conjugated double bonds or any similar elements.
Light absorption needs
charge carriers with some mobility, so they interact with light: free electrons, electrons whose orbitals can deform, ions or polarized molecules with some mobility like rotation or vibration... There are many ways!
Now, if you want to produce a dye, it must absorb some portion of
visible light. Though, electrons (the most efficient absorbers) tend to be on rather
stiff orbitals because these are small. The solution is to produce molecules with huge orbitals, which are easier to deform by the incoming light, so the resonant frequency (frequency band, more often) is lower and falls within the visible spectrum.
The most common method to obtain huge orbitals is to have
electrons delocalized over many atoms, typically by conjugated ethylenic bonds. Phenyl groups at the ends also contribute to orbital "elasticity" by sharing an electron lack or excess among many atoms.
Though, these tricks for visible light
aren't needed for UV when light frequency naturally matches the transitions between small orbitals.
More: for light absorption, the main molecule isn't necessarily the absorber. The slightest amount of impurity or dopant already colours a crystal; for instance quartz takes other names depending on tiny quantities of other components that change its colour.
Interesting for lasers for instance. Lasing needs a population inversion, meaning that >50% of lasing molecules are excited: this would destroy the lasing medium if pure. But in a YAG, the transparent garnet matrix holds the lasing ions, which are the active medium despite being in dopant quantity.