Choosing the character of Gogol to focus on the quest of identity is a good choice indeed. There are two ways to approach this topic with the character of Gogol: his search for cultural identity or his search for emotional identity.
Let's take Gogol's quest for cultural identity first. As the story opens, Gogol wants nothing to do with his Bengalese family. Gogol was born in America and feels tremendous pressure to be accepted as "American." Gogol's family is seen simply as his stumbling block at this point. It is Gogol's father's death that is the catalyst for this quest. After Gogol's father dies, Gogol suddenly realizes how important his own Bengalese cultural identity is.
They were things for which it was impossible to prepare but which one spent a lifetime looking back at, trying to accept, interpret, comprehend. Things that should never have happened, that seemed out of place and wrong, these were what prevailed, what endured, in the end.
He becomes less interested in the American Maxine and more interested in the Bengalese Moushumi. Gogol seems to find his fulfillment in his Bengalese culture through this marriage and treats his wife well. If the story ended there, it would certainly be a happy ending. Unfortunately, Moushumi has issues of her own.
Another way we can analyze the quest for identity is by focusing on Gogol's emotional identity. Especially at the beginning of the story, Gogol is very immature in dealing with his emotions. He is quite stiff and closed-off unless he is involved with a woman (such as Maxine) at which point he becomes obsessed and unable to think of anyone else. Just as in the first quest for identity, it is the death of Gogol's father that provides the impetus for change. After Gogol's father dies, Gogol realizes what is really important. Further, he matures enough to understand that Maxine is not necessarily the best woman for him. He is beginning to embrace his family's roots and beginning to allow his own emotions to be a source of pride. When this comes to fulfillment, he is able to meet, court, and wed Moushumi. He has learned how to be a good husband (even though Moushumi certainly does not want to be a good wife).