Crystal Healing

A week ago, I bought a rough gemstone from a local trader. I made my friend cut it into oval facet cut with european style. Now I have this beautiful but unnamed gemstone.

My friend told me that it's a color-change sapphire, but he argued with my other friend. He said it's an alexandrite. With they're argue like that, I have no idea what is the gemstone name actually.

The color is deep purple with red effect under sunlight. The more the yellow light, the more red it become. Under pure white light (like LED flashlight), the gemstone become bright blue.

Please anyone, help me identify this gemstone. I love colored gemstone but I'm no gemologist. Lab test is very expensive in my country.
As far as I know, Alexandrite have a green color. But my gemstone don't have any green color.