I think you need to take it to a jeweller first who can check whether it’s a natural stone or a synthetic by its refractive index. Other than that it needs a examined by a gem expert (using a microscope and various filters etc) to identify it, there are actually quite a few natural gems that display colour change including Garnets and sapphires. That said, an Alexandrite is expected to be green / blue tones in natural light and turning a purple / red colour under incandescent light ie candle light.