Although a majority of astronomers / cosmologists seem to favour a Big Bang model, there is a significant minority that does not accept that redshift is mainly related to cosmological distance and due to expansion of the Universe. This minority includes Arp, Narlikar, the Burbidges and others. And me. ;-)
So it would be sensible to consider a definition of redshift in terms of various components that include both sides and then argue about which components actually exist. We actually observe wavelengths or frequencies of light rather than redshifts, so that the terms generally have the form (1+z) which is the relative wavelength and the various components are multiplicative. By this I mean that if there is a gravitational redshift and a velocity redshift and a cosmological redshift then the three (1+z) components must be multiplied together to get the final (1+z) that is observed. So here is a list of various proposed redshifts and a notation for them:
z_g Gravitational redshift. This is well understood and not disputed at smaller scales although there might be factors that can be debated at large scales in connection with say missing mass and galaxy rotation curves. However this is not in dispute in the current proposal.
z_v Velocity redshift. By this I will mean only local peculiar velocity of motion, not expansion of the Universe. The Doppler equations are known and not disputed by either side.
z_c Cosmological redshift. By this is meant a redshift that is proportional to distance and results from expansion of the universe, or in the alternative model from changes of particle mass over time (Narlikar Variable Mass Hypothesis).
z_i Internal redshift. This has been proposed by Arp as an additional component which is not accepted by big bang cosmology. If z_i can be shown to exist then it undermines standard cosmology because redshift then has an origin that is not due to expansion but some other causes. To be sure that z_i is not zero requires proving the real association in 3D space of objects with very different redshifts that cannot be explained by gravitation or velocity. I suggest that z_i > .01 gets suspicious and z_i > .1 is indisputable proof.
z_o Observed redshift. It is then expressed as:
(1+z_o) = (1+z_g) * (1+z_v) * (1+z_c) * (1+z_i)
To summarize, the above formula would be accepted by both big bang and alternative cosmologies, except that z_c has a different interpretation as to cause and z_i is always 0 in big bang cosmology.
This is a rare case where the big bang has less parameters than alternatives. It is worth mentioning that having more parameters confers an advantage on a theory which has nothing to do with its merits (it can be called "curve fitting"). Therefore it is reasonable that a high standard of proof be required to accept the additional parameter. In particular it would be useful also if the alternative theory had other measured parameters that correlated with z_i so that a reasonable and coherent explanation for its existence is offered.
According to Arp, most galaxies only have small values for z_i and this is certainly required so that scatter diagrams of redshift versus various measures such brightness are found to be decently correlated. For galaxies we might agree that in general, z_g, z_v and z_i are all very small, say <.001 typically and certainly <.01 in the vast majority of cases.
However for quasars the alternative model proposes that z_i may be very large, often of the order of 1 or even more. Quite clearly such differences cannot possibly be accommodated within z_g or z_v and so demonstration of quasars with such deviant internal redshifts would disprove so-called standard cosmology. It is worth mentioning that there are classes of active galaxies that fall between these extremes and might have z_i of >.01 and so be difficult to reconcile with big bang, but this proposal will deal only with the more extreme class referred to as quasars or QSO.
I do not intend to get into what the definition of a quasar is, leaving that entirely to the astronomers. It is not relevant to the arguments offered here. Now to those arguments, and a new proposal that I shall put forward.
On my web site I have
a page which mentions this and I quote from there to give some references: (I will edit out some material)