These are most likely manganese dendrites. It is not a fossil, and not organic. These usually form in cracks in rocks, and most likely this slab was broken along an existing crack.
You can read more about it now: https://www.mindat.org/min-26645.html
One comment: this is not basalt. Basalt is black. Most likely some form of limestone.
They are quite pretty, but they are not fossils. They remind me of the patterns that frost sometimes makes on window panes. A similar process is at work here, though with mineral bearing water infiltration rather than frost being the cause. The water infiltration deposits crystals in the rock.