It's always hard to photograph gemstones up close, and especially hard to capture color shift! That said, it looks as if your stone has some minute bubbling and flow marks that would tend to indicate a synthetic (glass) stone rather than a mined alexandrite. Faux alexandrites have been around for a long, long time - including both glass varieties and lab-grown varieties. On the plus side, having a faux alex means you can wear it with impunity, and without having to worry about the possibility of losing or damaging a very valuable stone!
Yes to it being a synthetic alexandrite. They were hugely popular in the 1960 /70 s and weren’t “cheap”. A genuine alexandrite over 2 carats is rare, eye clean rarer still (they are like emeralds in terms of the natural crystalline flaws). Still it’s very beautiful so wear it and enjoy it.