What is the view that everything is ultimately composed of four fundamental elements called?
As Joe's comment alludes, this is definitely one branch of the aptly-named "elementalism."
As per my understanding, it holds that everything arises out of or is modeled after the four primary elements: Earth, Water, Air, and Fire. All of the higher, more complex structures that we have are assembled or expressed through these basic building blocks.
According to this site, the elements can be defined as follows:
Earth – The element of Earth represents the physicality and substance of our lives. The Earth evokes images of dirt, rocks, mountains and stones. At the human level, it implies a relationship to the physical foundations that underpin our existence: our bodies, our planet, the food we eat, the exercise we do… all of the everyday physical factors.
Water – The element of Water is linked to the mental aspects of our lives: the intellect, perception, sequential cognition, concentration emotions and memory. Where the body forms the basis for the element of earth, this collection of mental attributes forms the basis of Water.
Air – The context that surrounds us but which is normally invisible is our culture and our relationships with other sentient beings. Where the first elements seem to express a single being, Air is the connection between multiple beings. Politics, art, education and other group activities fall under the purview of Air.
Fire – The final primary element is Fire and that represents the actions that we take. Choices, decisions, practices, rituals… these are the realm of Fire. In this respect, it should be noted that Fire is an element that can be wielded both productively or destructively. A campfire can bring warmth or cook food – the same fire can burn down a house or destroy a village. Fire provides the essential feedback in the elemental process.
I assume that this derives from Aristotle's four elements of the same names, which he first enumerated in his treatise, On Generation and Corruption. However, these actually appear to originate from an even earlier thinker, Empedocles of Acragas. Of course, I've never seen any documentation confirming this lineage with respect to the more modern school of elementalism. Modern science, of course, has dismissed these early theories about the origins of matter and replaced them with a much more complex list of elements, taking a slight detour at medieval alchemy along the way.