Looking for and searching for is both translated as "keresni" to Hungarian. However, if you are searching for a doctor, you are opening every door and drawer to actively, phisically locate the doctor. :) The translation assumed it is not the case and you are simply asking around for a doctor.
Regardless, both should be accepted. Searching for is equally as correct and the inconsistency with which "looking for" and "searching for" are accepted and rejected here is eternally frustrating. I can't remember how many I've reported now.
hm ok, i assumed in this way they might be also used as equal. But this makes sense (;
Is it coincidence that "keresni" looks like an expanded version of "kérni", or are they related?
I am no linguist, but aside from a possible common root (which is also debated) there is no close connection.
Agreed that in English both "searching for" and "looking for" are acceptable translations.
The fact that there is a direct object is not enough to trigger the definite conjugation. The direct object must also be definite. That is, here it would have to be THE doctor rather than A doctor.