What is the ruling on being named simply "Al-Zahra," and is it better to be named "Fatima Al-Zahra" in full?

There is no objection to naming a girl "Al-Zahra'". This is a title given to Fatimah, the daughter of the Messenger of Allah, may Allah's peace and blessings be upon him, and it means the radiant, luminous one. Sheikh Bakr Abu Zayd mentioned that Al-Zahra' refers to a woman with a radiant face, one who is fair and illuminated. From this, the hadith in Surat Al-Baqarah and Al 'Imran states: "Al-Zahrawan," meaning "the two luminous ones." The name "Fatimah Al-Zahra'" is one of the compound names, and there is no objection to naming with it.