Emperor Fossil is a coelacanth and appears to be a variant of King Fossil.
In Darius Twin, he, along with Queen Fossil, battle together at Zone B and Zone C. Both have only one means of attack. In the Emperor's case, this is spitting out a rather large beam from its mouth. Like many of the Fossil bosses, its weak point is his mouth when open.
The two Fossils (pink for Queen, green for Emperor) simply move up and down the screen throughout this encounter, each Fossil taking turns to take you out. One fish starts its attack as the other reaches the bottom. Emperor Fossil fires big, Burst Out-like beams, while Queen Fossil floods the screen with slow-moving laser bursts (a proto-manic attack?). While it is trickier to dodge, you should look out for Emperor Fossil's straight laser as it is noticeably faster than the other attack. However, I wouldn't recommend taking him out first, as it can get hairier with just Queen Fossil on the screen.