Someone already answered really well regarding if the can be melted together. I have a possible idea for an alternative though. While it would not be amethyst and sapphire, there are natural ‘colour change’ gemstones that show a different colour depending on the lighting they are under (incandescent, fluorescent etc.)

A great example is alexandrite which is brain-numbingly expensive. But there is also natural colour-change spinel that changes between purple and blue (for your amethyst and sapphire colours). Spinel isn’t well known, is relatively cheap and has a similar hardness of sapphire, making it really good for everyday wear. It’s often found alongside sapphire in nature.

While alexandrite is typically greed and red, synthetic lab-grown alexandrites can be a blue and purple colour. I’ve seen the blue purple with flame-fusion synthetics (nothing to do with fusing gemstones together). Hydrothermals are much nicer but from memory are closer to green red. Synthetics are a great inexpensive option.

Source: worked in gemstone import industry for many years. I’m biased, but I’d recommend going through a registered jeweller if you want to go the natural gemstone route as buying stones online isn’t always going to get you what you asked for or the quality that you should be getting for your money. I know because I inevitably see these stones when customers bring them in and ask me what they’re worth.