as ex Auto Electrician, now retired..... let me explain the Alternator warning light and its circuits, The Blue wire is "Dash warning lamp", removing this wire and with Ignition on, then earthing the wire to metal will put the light on...check it does, when the engine is not running the Terminal in Alternator is "Negative" hence the light will be on!..once you start that terminal becomes "Positive" charging voltage...the Dash bulb [ 2 pin capless bulb ] is Ignition live on switching on [ ensure a good live 12v there ]...the other is from the Blue wire...hence you see the light must go out on running, something not right with yours, im thinking your new Alternator...does it have 2 small terminals ?...as some later Alternators do..one terminal was indeed W/Light..the other is AC output...will give your fault, but im thinking there is s Dash problem on what you write, but ive given you the circuit, so you should be able to check, bad earthing on Dash can give all sorts of feed backs, one other item that comes to mind, check the "Fuseable links" on the Battery Positive cables for the looms...they did "Rot away" as I remember back in the day...most got done away with...they give when Bad a heavy volt drop on Alternator main heavy Red live cable, put the W/light on Dim when running. hope of help. Peter