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Indeed. It is very interesting and I have worked with them for quite a while. I recently wrote a blog that was featured on Weights and Biases to demonstrate their usefulness. You can give it a read if you want but as of now, I have some disappointing news for you: The library mentioned in the paper uses different weights for real and imaginary parts which is expensive and forms a completely different loss landscape(as demonstrated in my article as well) so it's not similar to the original Theano implementation. I opened a PR on Tensorflow's GitHub as a starter for adding complex weight initializer support to TF but Francois outright said that they are not interested in pursuing complex valued networks as of now (here). So you shouldn't be surprised if you only a see a few improvements or research papers in this field in the coming years. Additionally, the point mentioned in the paper that it is not possible to properly implement layers like Dense and Convolution for complex variables is somewhat false. The default Keras implementation for Dense already supports complex variables and Convolutional layers can be implemented similar to the implementation at the bottom of this notebook. So it's not a matter of "unable to implement" but a matter of "who is desparate enough to implement it first" :)
CVNNs are quite under supported, there are several 3rd party libraries done for doing them. Here is mine: https://github.com/NEGU93/cvnn/