I set up my first tank when I freed my 2 Ecosphere shrimp this past October, and I first noticed this plant beginning to grow around that same time in a small 3-gallon Opae Ula tank I happened across in an aquarium shop here in San Diego. Turns out Mustafa had sold the shop those shrimp (and snails - and I assume he set up the tank for them as well) so that they could have them as part of their animal inventory. I've dropped by that shop periodically ever since for supplies and to check on that tank (have even bought a few snails and shrimp from it), and I noticed that this awesome-looking mystery plant was beginning to grow and spread throughout the tank. I asked a clerk there if Mustafa had intentionally placed that plant in there when the tank was set up, and he said no, that the plant just appeared out of nowhere one day and began to grow. (Btw, this tank has no macroalgae or any other foliage). If I recall correctly, they told me that Mustafa had been by since the plant began growing and saw it, so he is aware of it being in there (correct me I'm mistaken on any of this, Mustafa, if you see this post). I talked the guy I was speaking with into selling me a piece coral in the tank that had a nice little "grove" of the plant growing on it. At this point, the stuff was fairly spread throughout the tank, so he was ok with letting go of a cluster of it. Here's what it looked like about a month after I first brought it home this past March...Here's what it looks like now in my tank...
As can be seen, the original cluster has grown a lot in only about 4 months, and is beginning to spread throughout the tank. At first I thought it was encouraged to "root" in porous surfaces, hence the growth on the coral and other rough surfaces in the shop tank. But then I noticed it was actually rooting anywhere there was sufficient algae growth, whether the surface was porous or not, even on the smooth glass marbles I have in there. Here's a macro photo I took (using an Olloclip on my iPhone) of the base of one the stalks where you can see it growing out of the algae on the fairly smooth surface of a shell...
Here's a recent photo of the tank in the aquarium shop where the stuff has gone crazy recently...!
As you can see, each branch eventually "flowers" (see the little green "balls" in the above photos?), and then that branch subsequently falls over and dies. Here's a close up of a flowered branch beginning to die...
I was curious if anyone knew what this plant was, both the common and scientific name. I'm especially curious since it grew from seemingly nothing and it seems to flourish in brackish water. Assuming anyone is actually familiar with this plant, I also wanted to know if there is any danger of upsetting the balance of my tank in any way now that I've introduced it. The animals in the shop tank and my tank seem to be doing fine with it thus far, but I was wondering more about any potential long-term issues such as the plant using up all the nutrients in the water so that the tank is unable to sustain sufficient algae growth for the shrimp and snails to continue to have enough to eat. It does seem that my algae growth has dropped since I introduced it, but at the same time, my house has no AC or heat, and it has become a bit warm (often 80 to 84 degrees in the tank) with the onset of summer. So, that also could be affecting algae growth as well since plants (and the animals?) prefer relatively cooler water due to it being able to hold more oxygen than warm (i think - right?).
Thanks in advance for any info you might have...!
As mentioned, I got my sprig from that same tank at Aquatic Warehouse, and it definitely went nuts and flourished. And even though the first round of growth eventually started dying off after flowering (as many plants do), a new round of growth was already on it's way to replace it. I actually keep my salinity more on the low end of the brackish range - even after some significant evaporation, the salinity is around 1.010 (so even less when fully topped off). Here are some updates...
In any case, I've fixed your first post and I'll try to remember to look up again what this plant is. I did identify it at some point. I swear! 
I remember it took hours of research, too.