Could consciousness be a special kind of abstract object, such as a shape or a number?
For example, the squareness of a window depends on the window, but the squareness is presumably not independent of the form it takes (imagining a square requires imagining the form of the square). Could consciousness be similar where the “form” is just certain integrated physical processes?
Short answer
Yes. Naturalism does not pre-constrain theories about what "might be the case" and the idea that consciousness is identical to an abstract object is very much in the realm of logical possibility.
Preamble to longer answer
As several other commentators and answers have noted, there are aspects of consciousness (immediacy, time-based, self-identity, action/agency) that suggest that consciousness has incompatible properties with abstractions, such that abstraction/consciousness identity seems extremely implausible.
This is also the case for consciousness and material, and these strong differences are why materialist identity theories are also intuitively rejected by property dualists.
These intuitively identified differences become a rationale to adopt a "substance triplism" model of our universe, as Frege and Popper did.
An intuitive problem for dualism, or triplism, however, is our further intuition that fundamentally different things cannot interact.
Yet our world seems to have regular interaction between these sorts of radically different things -- abstractions are embedded in physics, and our mental reasoning, consciousness is key to what we do and why, etc.
So any dualism or triplism that accepts the interactive causation we observe, must admit that these three types of things share enough "essence" to be interactive, and the absolutist non-interaction of our intuition is wrong. This possibility has been strongly reinforced by the very non-intuitive features of modern physics (QM and GR). Our intuitions, forged in dealing with medium-scale life as hunter-gatherer animals, are not a valid guide to the more fundametnal nature of our universe.
Accepting this opens up the logic space of possibilities, such that all of materialist property dualism (matter has both material and consciousness properties, that it exhibits in parallel), materialist emergence of consciousness, materialist identity theory, mental idealist emergence of consciousness, mental idealist identity theory, and abstraction emergence followed by abstract identity theory would then all be POSSIBLE. And then we would need to examine how well they match the world to see if the supporting evidence for them is strong enough to overturn our intuitions.
The last, that functions are emergent from brain processing, then functionalism develops either dual properties, or is identical to consciousness, is basically what functionalism and its descendant ideas discussed below proposes. So now on to:
Longer Answer
This has been repeatedly proposed. Functionalism, which for a time was the dominant "physicalist" view in philosophy of mind, is the thesis that that consciousness is identical to some particular function. Functions are abstract objects, so this is an abstraction identity theory.
Computationalism, the thesis that consciousness is computation, is a subset of functional identity theory, where the function proposed to be conscious is a computation function, an algorithm. AI theory has been operating under Algorithmic Identity Theory for most of its history. The idea of "digital uploads" and of "simulation theory" both implicitly rely upon this assumption.
Falsification testing
The problem for functional identity theories is similar to the problem for neural identity theories -- we have test cases where we are conscious without doing a function (experience qualia for example) and cases where we or our machines do functions but are not conscious (solve a math problem unconsciously, then suddenly intuit the solution consciously, for example).
Most functional identity thinking has involved proposing special case functions that might be identical to consciousness, rather than all functions.
Examples of special case functions proposed
For the first several decades of AI, it was assumed that consciousness would require both doing computation, AND some minimal throughput processing speed after which consciousness would suddenly emerge. After our machine throughput increased by many orders of magnitude, with no emergent consciousness, this idea has mostly now been abandoned.
Generic "complexity" of processing was also postulated as being a possible emergence trigger for sudden appearance of consciousness. However, 99+% of our processing is unconscious, and it is run on an equally complex computational network as any conscious processing could use, so this is not particularly plausible based on observed neuro-processing.
It was also proposed that neural net processing is conscious, IFF it is recursive with feedback and feed forward loops. This proposal was to try to explain why our computers are not conscious, while we are. This too runs into the problem that 100% of our brain processing is recursive neural net processing with feedback and feed forward loops, but 99+% of it is not conscious.
Three theories about the structure of algorithms that are needed for consciousness have been:
All three of these suffer from our neural net structure, where we don't seem to even have "algorithms" in our brains. Also, most qualia are not self-referencing, or strange loops, and the daydreaming and very peculiar random walk of our "stream of consciousness" seem to violate global workspace theory.
Integrated Information Theory holds that only algorithms that are "run" on a "high Phi" material substrate are identical to consciousness. Phi is a measure of the uniqueness of different aspects of a processing substrate, and our brains have high Phi, while our computers have very low Phi. IIT is therefore an effort to explain why our computers are not conscious when running algorithms that we run. IIT once more suffers from our unconscious processing also occurring on the same high Phi substrate that it proposes our consciousness to run on.
Conclusion
Your idea has been widely explored, and is involved in most of the physicalist efforts to explain consciousness over the last 3/4 of a century. These ideas, however, are among the suite of ideas that are considered to not have succeeded in addressing the Hard Problem of Consciousness over that time.
Consciousness can denote the I-sense, or it can denote the quality of having the I-sense, or it can denote the content or qualia of the I-sense. Qualities are abstract, but senses and sensory experiences are axiomatically concrete.
Taking 'consciousness' as understanding in the follows sense
¶ 31 . Being-there as Understanding
State-of-mind is one of the existential structures in which the Being of the 'there' maintains itself. Equiprimordial with it in constituting this Being is understanding. A state-of-mind always has its understanding, even if it merely keeps it suppressed. Understanding always has its mood. If we Interpret understanding as a fundamental existentiale, this indicates that this phenomenon is conceived as a basic mode of Dasein's Being. GA2 p.182
the problem is that Being cannot be an entity. It cannot be an object.
‘Being’ cannot indeed be conceived as an entity; enti nun additur aliqua natura: nor can it acquire such a character as to have the term “entity” applied ‘to it. GA2 p.23
Because if existing things are founded by existence—in other words, if beings come from being—then being cannot be a being. (i.e. a thing can't be the foundational cause of things, because then what would cause that thing.) So Being is not a thing and consciousness is not an object.
On the other hand consciousness is tangible. – But that's kind of secondary effect. The origin isn't. The wellspring of thought is intangible, invisible, although the thoughts produced are tangible.
