Dirac equation is always written with indices. Is there any way to write it down without any indices ABSTRACT or not, and without coordinates,basis vectors etc..?
Writing out $\gamma^\mu \partial_\mu$ is really no different than writing anything that looks like an inner product (for example, the Laplacian). If you're concerned that the indices don't make the frame-independence obvious, then just write it as a four-vector dot product: $$ (i\gamma\cdot\partial - m)\psi = 0. $$
I'm not saying that this notation is usual, or even that you will find it anywhere, but it seems no different to me than writing, say, $\nabla \times \nabla f = 0$ or $\nabla \cdot \nabla \phi = -\rho$.