Yes, I've often thought about this too. The argument from complexity is backwards. It is simplicity, not complexity, that is the identifying feature of design. Just look at all of the designed objects around you. What are the features that make these things look designed? More than anything, it's that they are assemblages of simple shapes. Straight, often parallel, lines; flat, and often very smooth, surfaces; near perfect 90degree angles; circles; cylinders; spheres; etc. Even the "irregular" shapes in designed objects usually have some kind of easily identifiable order, such as smooth curves or some kind of symmetry. And when you think about some of the more complex designed objects, things that are functioning systems with many parts, you still see that distinctive designed simplicity in their elements. The LCD monitor that I'm looking at right now is, overall, fairly complex. But its the simple elements that make it appear designed. The very flat and almost perfectly rectangular display surface; the simple, smooth, symmetrical curves of its housing and stand. And if I were to look at the more complex internals, the simplicity of design would still be apparent. Flat circuit boards. rectangular components in orderly arrangements. Uniform lines, etc.
When we see such simple forms in nature, it is invariably the result of natural forces or processes that produce those forms without any intelligent guidance. Crystal formation for example, or the tendency for gravity to produce spherical objects in elliptical orbits. Beyond that, nature tends to produce very chaotic and complex forms, quite different from things that are designed. Very few straight lines, flat surfaces, right angles, accurate circles, smooth curves, parallel lines, etc.