Why there is only one type of artificial neuron? I find it strange that so many deep learning tricks and improvements have been invented in the past decade but I never heard about someone trying out different models of the artificial neuron other than the one based on perceptron:
y = w*x + b

I haven't been able to find much info on this online which is surprising. I am not a machine learning expert but from the little I know, it would make sense to me to at least experiment with other options. A trivial example, what would happen if there was a layer in a network consisting of neurons where
y = w*x^2 + b

Maybe there is an obvious answer to why my suggestion above isn't a good idea but I would prefer an answer that explains why generally this is not being looked into (at least as far as i know).
The example you cited (using x^2 instead of x) is the idea more popular outside deep learning community, called feature engineering. The trend in neural network modeling is instead to,

Play with weights (w) and fine tune them.
Not change the input vector (x) but feed it to the network directly.
If a single layer neural network is not good enough, add more layers.
Introduce non-linearities using activation functions.
And in general, not hand-roll features (like x^2) but let neural networks discover such features.