How easy or hard is it to write a new "back end" for Eigen? We have a custom-designed vector processor and would like the same application code that runs on off-the-shelf computers to run efficient on this special machine.

It's quite straightforward, you basically have to copy-paste the PacketMath.h file in Core/src/arch/SSE, and adapt the p* functions.

BTW, what is the status of AVX2 support in Eigen? This is something we'd be very excited to see!

as far as I known nobody started working on it, but since we already have an AVX backend, adding AVX2 support for integers should be rather straightforward.

(dropping AVX2, now asking about AVX1)

In version 3.2.2, I see no mention of AVX in Eigen/src/Core/arch/SSE.

In your repository, I see you have an Eigen/src/Core/arch/AVX directory, which is not included in the 3.2 branch.

If I want to get some AVX optimizations with maximum overall stability, what would you recommend? What would it take to merge arch/AVX into 3.2?

The current devel branch is pretty stable and I'd recommend to use it. However, keep the current revision as we'll merge a huge refactoring of the code soon.