Yes, we can choose an element other than carbon-12 for atomic mass reference. The choice of the reference element is arbitrary, and it depends on historical conventions and convenience.
In the past, the atomic mass scale was defined relative to the isotope carbon-12, which was assigned a mass of exactly 12 atomic mass units (amu). However, in more recent times, the International Union of Pure and Applied Chemistry (IUPAC) adopted a new standard for atomic masses based on a different isotope of carbon, carbon-12 being only one of the possible choices.
The current standard adopted by IUPAC is based on a weighted average of the atomic masses of all naturally occurring isotopes of an element relative to 1/12th the mass of carbon-12. The atomic masses of elements are now expressed in atomic mass units (u) or unified atomic mass units (daltons), and they are determined experimentally using sophisticated mass spectrometry techniques.
So, while carbon-12 is commonly used as a reference for atomic mass, it is not the only choice, and other elements could be used as a reference point if desired.