To be 100% sure one would need to study a thin section of the rock under a polarizing microscope. That said, however, I am 99.9% sure that this is a vesicular basalt in which large bubbles of volcanic gas have been frozen in as the lava rapidly solidified. I think your assumption that it is basalt is well founded. Under the microscope you would most likely find a mass of tiny interlocking crystals. The white laths would be plagioclase feldspar, whist the brownish / greenish or possibly grey crystals would be various types of pyroxenes. In most basalts these are two small to resolve with the naked eye. In addition there is likely interstitial glass, which froze before it had time to crystallize, and some opaque grains of magnetite. Some varieties of basalt will also have traces of other minerals.
The white and brown minerals are late stage precipitates introduced by heavily mineralized ground-waters (common in volcanic environments). You can clearly see the hairline fractures through which the groundwater seeped, probably under considerable pressure. Note that if a bubble is not intersected by a hairline fracture then it is hydraulically isolated, in which case the bubble remains empty - no mineral deposition. You can see a few such examples in the photo.
If the white crystals do not effervesce with acid, and are hard to scratch then they are almost certainly quartz. Otherwise they are most likely carbonates such as calcite, aragonite or dolomite (much more rarely hydroxy-carbonates).
Iceland is well known for a range of zeolites (complex hydroxy-alumino-silicates) which characteristically infill the vesicles (gas bubbles). Most zeolites are rather more coarsely crystalline than the large pinkish precipitate, but one could not rule zeolites out as a possibility.
As for cutting it in two, you would see more of the same. The larger vesicles / bubbles are more likely to intersect a fracture and be filled by any of the above late stage precipitates. There will be many smaller bubbles. If you carefully polish the cut surface, and examine it carefully with a hand lens you should be able to see the component crystals of the basalt.