I am not a mathematician but I am an artist and fractal enthusiast who spends lots of time creating art using this geometry. I recently discovered online you can make fractals in MS Paint (there's plenty of info online about this) and one of the first fractals I created was the Viscek fractal. First I did the cross form, then the saltire form, but then I had an idea: why not combine the two? So first I start with a square, then add a square to each side like you do with the cross form. Next, I duplicate this new cross shape and add it to the corners of the cross, creating somewhat of a tile pattern.
I repeated this for an unknown amount of iterations until I eventually ended up with a stunning final result.
Surely a shape this simple must have been discovered before, right? I doubt an amateur like me was the first one to think of this. Either way, I still think it's neat I came up with it on my own even if I wasn't the first.
Did you just invent a new fractal?
Probably. The process you are using can generate infinitely many different ones, so if you start with anything other than the simplest initial pattern you are likely to build a fractal no one has ever seen before.
This one is compelling. Elegant. Stunning, as you say. I think you should make one in cross stitch. That does not make it particularly interesting as mathematics, whatever it is as art. But don't let that discourage you. Keep playing.