Religionists often say that things like the paintings of Rembrandt; Beethoven’s symphonies and theatrical masterpieces from Shakespeare are examples of both design and complexity. While I will agree that design is inherent in any work of humans, I have to dispute the complexity that is claimed. These are, in fact, NOT complex. I would argue that the reason they (and all human “designs”) are so noticeably not “natural” is because they are simple and non-complex. Allow me to explain.
Consider a radio. When you tune through the frequencies, you hit lots and lots of static and noise (I’m not talking about the talk-radio hosts). However, every so often, you tune in an actual broadcast station, and then you hear music, commercials, talk, etc. If you compare the complexity of the signal between the static/noise of a non-broadcast channel with the clearly tuned broadcast station, you will see that the broadcast station radio wave is simpler, far less complex, and far more regular than the non-broadcast channel. This is because in order to communicate, we have to use regular, simplified and more pure (i.e., less complex) sounds and frequencies.
Likewise, a painting is also a less complex, simplified application of paint than a completely random splattering of the same amount of colors on a canvas (apologies to Jackson Pollack). It is this appeal to the simplified, less complex, more structured features that humans equate design. And as such, your examples actually appear to show exactly the opposite of what you intended. It's not that a Beethoven symphony, Rembrandt painting or Shakespearean play isn't more "designed" than what a typical gradeschooler could produce, but that the gradeschooler could easily produce a “musical score” that contained more complexity than Beethoven, a picture that was more random (and thus more complex) than Rembrandt, and a string of letters that contains far more complex ordering than anything Shakespeare wrote.
Humans hate “kludges”, those haphazard workarounds that use ill-suited parts to jerry-rig a system to do something it wasn’t originally intended for. Rube Goldberg-esque contraptions (http://en.wikipedia.org/wiki/Rube_Goldberg ) don’t appear to be good designs, because they are inefficient, awkward, prone to failure and completely impractical. However, when you examine them, you see that they are far more complex than what any “normal” person would design to do the task. Redundancy, excess steps, using ill-fitting parts to do a job, etc are not what makes a good, efficient, clean design. Take a look at a modern computer -- it's got lots of parts, but few redundancies (only one CPU, only one power supply, only one hard drive, only one way for the wiring to connect, etc). It's well designed by humans, but it in no way resembles a cell. A cell has multiple "CPUs" (chromosomes -- humans have 23 pairs), dozens of "power supplies" (mitochondria), multiple "hard drives" (in eukaryotes, there are at least two copies of each gene, and often hundreds to thousands of copies of each protein), plus the flexibility to be able to access just about any of those genes simultaneously using multiple mechanisms and pathways. And that's just comparing a simplified example of a cell!
When we look at the natural world, we actually see far more of Rube Goldberg than Beethoven, more kludges than Rembrandts, and more haphazard randomness than Shakespeare. It’s not that humans can't produce complexity, but that in searching for patterns, structure and organization, we strive to simplify, consolidate and reduce redundancy – all of which are the OPPOSITE of complexity and the randomness in the universe. In short, complexity isn’t the hallmark of human design, simplicity is.