In intelligent design theory, what exactly are the ID scientists comparing life to, to determine its complexity?

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My understanding is that they're not really comparing it to anything. The idea is that the structure of DNA is, in itself, so complex that it could not have been produced by the kinds of processes postulated in the theory of evolution. There are ways of measuring complexity in such cases, or at least there are ways of trying to do so, but it is extremely difficult to provide a good account of this kind of complexity. A large part of the reason is that DNA is finite, and most of the mathematics relevant to the study of complexity counts everything finite as supremely simple. Still, there are ways one can go here (using, for exmaple, the resources of information theory). But part of the criticism of many arguments by proponents of intelligent design is that they operate with inadequate accounts of complexity.

The more fundamental criticism of these arguments, though, or so I take it, is that there simply isn't any remotely plausible argument that the structure of DNA is too complex to be produced by evolutionary processes. Of course, such arguments have been offered, but they have not been at all convincing to the great majority of working scientists. Indeed, scientists who have not accepted those arguments have not found anything of interest in them, and that is important: Scientists don't just respond "yes" or "no" in these kinds of cases; they can respond, "well, no, or probably not, but this bit seems interesting". That hasn't happened. Scientists react in pretty much the same way mathematicians do when someone submits a new proof that it is possible to trisect and angle with ruler and compass. We have overwhelming evidence that the proof is wrong. It can be amusing to see just where the proof goes wrong, but there's not really much illumination that is likely to come from the exercise.

An even more fundamental criticism is that, even if that were so, it's not clear why we should assume that an "intelligence" or some sort was responsible. David Hume long ago dismantled that sort of argument in his Dialogues Concerning Natural Religion.

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