"Socrates was searching for absolute truth."
Translation:Ο Σωκράτης αναζητούσε την απόλυτη αλήθεια.
Can we have για after αναζητούσε ?
Yes, if you want to show intention, e.g. I search in order to find. Same for the verb ψάχνω (ψάχνω για να βρω). In that case you can also follow with γιατί/επειδή (αναζητώ/ψάχνω γιατί θέλω να βρω). Για and γιατί are closely related but note that the sentences have to be structured differently, with the subjunctive and a secondary clause, respectively.
Otherwise, a direct object in the accusative follows the verb straight away.
can we have "έψαχνε" without "για" right after it?
Yes. I'm not even sure if both structures (ψάχνω with or without για) are correct but they are used all the same...
Mostly yes, but not necessarily. Both can be used for material or abstract objects. Αναζητώ means "calling for something persistently, searching persistently, or desire intensely" so it is more emphatic. Usually, in phrases like the above one, αναζητώ is used. Ψάχνω sounds really non-imaginative in sentences like that, and less intense.
Thank you troll. I suppose it'd be a bit like using "quest" instead of "search" in English.