"I have looked for you."
Translation:Ich habe nach dir gesucht.
can someone explain what function "nach" has in this sentence? for instance, why wouldn't you say "ich habe dir gesucht"?
It has the same function as "for" in the English, but it can be omitted. In your example, that would be "Ich habe dich gesucht".
Note that the case changes if you omit "nach".
Ich habe nach dir gesucht. dative
Ich habe dich gesucht. accusative
Remembering that "nach" has the meaning "after/towards", I would translate the sentence presented as "I have sought after you."
I put 'Ich habe für Sie gesucht' which was accepted, is it good German? Danke.
I am not a native speaker, but I would say your translation means rather "I have searched on your behalf", as he should have done it, but you did it for him.