This issue is mainly for tracking the upstream counterpart (numpy/numpy#14753) for compatibility purposes. However, there are unique challenges and opportunities in CUDA that makes this support worth considering independently of upstream (for the moment).

Specifically, many scientific computing problems may have a portion (could be a significant one) in the computation that has high tolerance of low precision. For these applications, using half complex could further save both memory footprint and computing time. However, AFAIK all CUDA libraries (cuFFT, cuSPARSE, cuSOLVER, etc) do not support half complex, so in CuPy a type cast to single complex is a must if CUDA libraries are to be used. Using elementwise/reduction kernels is free of such a problem, though, so it'd be interesting to target them as the first step.

If I were to work on this, I would start from investigating whether thrust::complex can support complex<__half> or not, and build up the infrastructure gradually.

cc: @smarkesini