The Amazon basin is one (or two if you want to split the northern end / Orinoco Basin from the western end) of only a 8-10 epicenters for agriculture in the world. Beginning around 8000 BP, Amazonian peoples began domesticating such notable crops as sweet potatoes, cassava, cotton, yams, cacao, tobacco, pineapple, rocoto pepper, and peanuts. From here, these crops spread throughout most of South America and north into Central America and Mesoamerica, with a few making it ever further. In the case of sweet potatoes, they'd eventually make it all the way to New Zealand before Europeans arrived. Most pottery in the Americas ultimately traces its ancestry back to the Amazon as well.

Aroudn 450 BCE, the people of the Amazon made another major agricultural innovation - terra preta. This is a man-made soil that vastly increases the fertility of rainforest soil and the longevity of farmland in tropical environments. It's also self-perpetuating, as it creates a micro-ecosystem that slowly converts more soil into more terra preta under the right conditions. Unfortunately, the techniques for producing this material have been lost and are still under investigation by modern soil science. In the meantime, we have to make due harvesting small amounts of terra preta from the sites that are regenerative.

Pockets of terra preta can be found throughout the Amazon today, and it's not surprising that where it is most abundant we also see archaeologically evidence for large-scale societies.

Many of these large Amazonian societies existed when Europeans arrived. The expedition led by the conquistador Orellana in the 1540s from Andes to the mouth of the Amazon provides our earliest historical look. For example, he encountered the Omagua who controlled a 300-500 mile stretch of the central Amazon. One Omagua town had an estimated 8000 people, while another stretched along the Amazon for 20 continuous miles, and a third had a warehouse full of ceramics waiting to be sent out for trade. As large as Omagua was, the people along the Amazon agreed that the Ica to the north were even larger and stronger. Orellana didn't encounter the Ica and specifically avoided going too far away from the river (he didn't want to split his forces or commit to an overland expedition), but he did chance upon the roads the Ica built through the forest. Further downstream, the vicinity of modern Santarém also densely populated by the Tapajós, who were said to be able to dispatch a force of some 60,000 archers in the early post-Columbian period. Archaeologically, this is one of the regions were terra preta is at its most abundant. Anna Roosevelt, an archaeologist who specializes in the pre-Columbian societies of the lower Amazon, described the region around Santarem as "of urban scale and complexity," and as part of a highly stratified tributary political system. At the time of Orellana, his contacts in the area said they were led by a queen called Conori who ruled from further up the Tapajós River. Though Orellana never met Conori, it was after this warrior-queen that the Amazon was named. On Marajo Island, at the mouth of the Amazon, there was another large Amazonian society. The largest community there, now the archaeological site known as Camutins, had a population of an estimated 10,000 people. Camutins had declined prior to European contact, but the Marajoara were still numerous on the island and neighboring areas, and divided into several smaller competing polities which resisted Portuguese influence until the mid-1600s (when they were known as the Nheengaibas, an exonym applied to them by the Tupi).

Ultimately, these large Amazonian polities were highly disrupted by European diseases and warfare with Europeans and other Amazonians. In this turmoil, the Omagua refugees headed upriver to join with others into a coalescent Cambeba identity. The Marajoara were eventually defeated by a combination of Portuguese invasions from the south and Carib invasions from the north. The Tapajós identity faded away due to a combination of emigration to other communities in the Tapajós and Xingu River valleys, assimilation efforts by Jesuit missionaries, and warfare with neighboring peoples like the Wuy Jugu.

Unfortunately, I don't know of an English translation of the chronicle of Orellana's expedition, but River of Darkness is a good substitute if you want a more detailed look into what the earliest Europeans in the area saw. You'll probably also be interested in Handbook of South American Archaeology, which will have the basic archaeological information in a single convenient location.

There seems to be an ongoing re-evaluation of the Amazon as a site for human civilization. A chronicler traveling with Francisco de Orellana, the first European to navigate the Amazon River, depicted the river as containing many densely populated and prosperous villages and even cities. Some of the stories compiled by Orellana's chroniclers were clearly fantastic like that of topless women warriors and this did much to discredit the account of Orellana's voyage.

Archaeology has somewhat vindicated Orellana's account. Clearing of the Amazon for farming and cattle grazing has uncovered manmade [geoglyphs] (http://www.nytimes.com/2012/01/15/world/americas/land-carvings-attest-to-amazons-lost-world.html?_r=2) that are 1000 to 2000 years old. The Amazon forests may have been extensively modified for human inhabitation as well; for instance, archaeologists have discovered fishing weirs and complex series of dikes for water control in Bolivia. Excavations in the Upper Xingu region of Brazil also revealed 19 villages linked to each other by a series of roads. I don't know if that constitutes as a mark of a complex civilization but it does show that the people of these regions were organized enough to raise labor and resources for large scale public works projects.

The dense forests of the Amazon River did indeed complicate farming. The soils of these forests are poor and so detrimental to agriculture. Farmers could engage in cutting down the parts of the forest and burning the chaff to create temporary agricultural plots (with the ash of the burnt plant life providing the nutrients for the crops) or they used artificially created enriched soil, terra preta, for their needs. The terra preta had charcoal and pottery shards mixed in and is a good indicator of civilization.

Work is still ongoing and I've only touched on what I know. While archaeology is hampered due to the inhospitable nature of the terrain, technology like Lidar helps reveal sites covered by the forest canopy. It'll be interesting to see what will be discovered.

Sources:

Mann, Charles C. 1491. Vintage books.

Wade, Lizzie. Searching for the Amazon's Hidden Civilizations. Science. January 7th 2014.

Heckenberger, Michael C. et al. Amazonia 1492: Pristine Forest or Cultural Parkland?. Science. 19 Sep 2003: Vol. 301, Issue 5640, pp. 1710-1714

Edit: Corrected terra preta spelling error

Let's ignore Diamond for now, because his ideas are wholly inapplicable in the present scope.

Archaeology in the Amazon basin is a relatively recent thing. It's only in the past decade that deforestation and aerial surveying have revealed some of the many site features that have been covered by the dense forests for centuries, such as these at Fazenda Colorada. The Amazon basin as a whole is known to have held relatively dense populations with agriculture (including perhaps fish farming), organized labor forces, and social stratification. Many archaeologists would argue these are the markings of a "complex" society.

What constitutes a "complex" civilization in your question?

Is there any written record of when the Europeans discovered that the diseases they carried were decimating the native population?