Murkowski never ran as an independent.
From Wikipedia:
“Murkowski ran for and won a full term in 2004. After losing the 2010 Republican primary to Tea Party candidate Joe Miller, Murkowski ran as a write-in candidate and defeated both Miller and Democrat Scott McAdams in the general election. She is the second U.S. senator (after Strom Thurmond in 1954) to be elected by write-in vote. Murkowski was elected to a third term in 2016, running as a Republican.”
Write in candidate= independent.