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.
And no where in there does it say she quit being a Republican and ran as an independent.
She ran as a write-in, Republican candidate. Alaska law at the time allowed that for write-in candidates to still claim a party even without being the nominee. The list of write-in candidates that could be requested at the time even listed her as a Republican.