Bromance: as old as recorded history. (And we thought we'd invented it.) At the core of The Epic of Gilgamesh is the story of the powerful friendship between Gilgamesh and Enkidu. At the beginning of the story, Gilgamesh is a man possessed. It's hard to look at his relentless drive and not see it as an attempt by Gilgamesh to fill a void in himself. This, at least, seems to be the interpretation of the goddess Aruru—we think she's right. Along comes Enkidu and suddenly Gilgamesh stops acting like such a jerk. Apparently, all it took was the creation of a part man-part beast creature to help Gilgamesh realize that no man is an island.
Enkidu is the "rational" part of the friendship; once he is gone, it is not necessarily his death, but his absence and inability to counsel Gilgamesh, that leads Gilgamesh to go off on such an irrational quest.
When Gilgamesh goes from lamenting the death of his friend to lamenting his own future death, his feelings for his friend don't diminish. Gilgamesh and Enkidu are so close that, when Enkidu dies, Gilgamesh is basically losing a part of himself anyway.