Hi Elif,
I know dr. Nega, she (also) works at the Róbert Károly private hospital (
http://www.robertkorhaz.hu/en), where she runs the gynocological department. She is also a popular choice for expectant mothers. I went to her for one of my check-ups, when my own doctor was unavailable, and liked her a lot, even considered transferring to her when my own doctor pulled the disappearing act for a second time. She was thorough, really looked at all my paperwork, examined me with a highres ultrasound properly. This I thought was very good. She a loud, cheerful type of person. She is originally from Ethiopia, I think, but speaks great Hungarian and English.
She told me that my baby was very large for his age and that there was a lot of amniotic fluid - these things worried me, and I asked whether she thought these were problems, and whether she thought I would be able to give birth naturally. She reassured me that there was no reason to worry. Still, because she had said these things (exclaiming "that's a very big baby!" several times during the ultrasound), I remained slightly worried. I think this story may help you decide whether she is the doctor for you - I think she's very thorough and will answer questions in a helpful and full way, but make sure to ask if anything she says worries you. All in all I had a very positive experience with her.
I don't know her stance on types of giving birth (eg. whether she is a fan of natural childbirth, whether she has strong opinions on pain relief or positions or whatever), since I only met her properly once.
Incidentally, I also recommend the Róbert Károly hospital, where I gave birth. They have a wide range of services and possibilities and within the market of private hospital birthing they are actually on the cheaper side, I believe (although if your insurance is paying I suppose it doesn't make a whole lot of difference), compared with say Maternity in the second district. They also gave great care in the days after birth, with lactation consultants on hand etc.
With regard to hopitals, however, in the case of twins it's important to bear in mind that private hospitals (and also "normal" hospitals) will only "take you" after the 37th week is completed, which may not happen with twins. This is a law in Hungary - if you give birth before that, you will be taken to a hospital which has a PIC (perinatal intensive care) facility, as the babies are considered premature. This means that your chosen doctor will not be on hand for the birth itself, however, it still matters who you choose for prenatal care. Other moms, please correct me if I'm wrong in any of these things, this is just what I have heard.
Best, Aliz