Alexandrite occurs in a wide range of colors, but color change resembling fine emerald (green) and ruby (red), depending on the light source is so rare, that most people have never seen one. However, even lower quality alexandrite should change colors from khaki or yellowish green or green in daylight moving to brownish, purple, reddish-purple or red under incandescent light. Mixed light should produce either a grayer color or flashes of the green, purple, and sometimes red or yellow.
Good quality alexandrite is teal, emerald, or blue-green under natural daylight or fluorescent lighting that ideally changes to purple-red or red under regular incandescent light. Often a slightly grey, blue/violet/purple is seen.
If your stone changes color from purple in daylight to greenish-blue under incandescent light, then instead of genuine alexandrite, you own either a synthetic stone or some other gemstone variety which changes color. It could be sapphire or spinel, although it's very unlikely. I would recommend obtaining a gemstone identification certificate for your gemstone from a reputable laboratory. The most well known gemological laboratories are listed at: Where can I test and appraise my alexandrite?