Scalar fields have dimension 1, spinor fields dimension 3/2, and vector bosons like the photon dimension 1. According to the principles of renormalizability (along with others), this restricts the possible interactions a field can participate in, by restricting the possible combinations of fields in a Lagrangian. However, if we had a dimensionless field, we could have a kinetic term with four derivatives, and every combination of that field could exist with a coupling constant of dimension 4.
I can see one problem in how this field might couple to other fields. I just don't have the depth of understanding to see where the problems lead.