I have to tell you the first glance, I thought I was looking at the freshwater Cabomba. I honestly don't know the name of it if it's indeed marine plant.
I was looking through Baensch Marine Atlas and saw the pic very similar to yours. It indicates it's the young shoots of Caulerpa paspaloides. As the plant matures, each division will thicken up.
That doesn't look like Caulerpa paspaloides, though under very low light it MIGHT take such a form, can you back up a bit so I can get a sense of the scale? Judging by the grape of Caulerpa lentillifera on the left, I'd guess your specimen is an unusually large Caulerpa verticillata.
I attempted to take a few better shots but this stuff is against the glass and the grape is crowding around it. I found some pics though for comparison shots...
C. Paspaloides var laxa
C. Verticillata
The one in my tank kind of resembles both in some form or another. Like the Paspaloides it has many "sprouts" on each stem(unlike verticillata), but like the Verticillata is has many more stems.
I'll try to take more pictures tonight when the light from the window behind me isn't creating any glare on the glass.
The other two are about 200k each. Nice and big so they can be seen easier... I'll link to them directly to save a little bandwidth for those on a slow connection.