Photoshop natively supports HDR for 32-bit documents. However, 32-bit mode is missing several important tools, such as:
Black & white, Selective color, and Color balance adjustments
Healing brush, spot healing, patch, and content-aware move tools
Sponge, dodge, and burn tools
Adaptive Wide Angle
Dust & scratches filter
Smart Sharpen (which is technically available in 32-bits, but clips HDR values)
Reduce Noise filter (the Photoshop version, you can use the Camera RAW version on 32-bit images)
Workflows / actions such as frequency separation
Lumenzia offers an option to create 16-bit HDR images so that you may use these tools. This feature is invoked when you are working on a 32-bit image and click the selective color or color balance icon in Lumenzia (if you wish to convert for other reasons, just click one of these and delete the unnecessary layer). Note that this feature requires Adobe After Effects.
This conversion is done by encoding the image using a PQ transfer function instead of a standard gamma. This means that the 0.0 - 1.0 range of 16-bit values is mapped to 0 - 10,000 nits (which would be 0.0 - 100.0 in 32-bit values). This enables HDR image display in any bit-depth, but has several important considerations:
Your existing 32-bit work will be converted to a Smart Object. This may complicate subsequent editing and so this conversion is probably best done toward the end of your workflow.
Just like when working with profiles and unusual gammas (such as 1.0 / linear), numeric values will be quite different. This means that tools such as curves will work much differently than normal.
The encoding is such that true SDR white will show an RGB value around 121 (roughly 0.48 if measuring with 32-bit values). The rest of the encoding range are mapped to HDR values. An RGB value of 255 (1.0 in 32-bit) would map to 100.0 in 32-bit measurements (ie roughly 6.5 stops over SDR white).
You should generally avoid final output values above RGB 190 (0.75 in 32-bit measurements), as this maps to extremely bright HDR values which would be clipped on most displays.
While Photoshop will render this file correctly, not all software supports more advanced ICC profiles. This includes ACR and Nik Color Efex Pro. You may convert back to 32-bit to work around this. You may do by that by select all layers of your 16-bit HDR, converting to a Smart Object, and then using Image / Mode / 32 bits per channel. Or you may simply flatten the image and convert to 32-bits.
Note that you will likely need to convert back to 32-bits for any HDR export to ensure proper viewing in web browsers and other software.
You can technically convert this document to an 8-bit HDR. That creates a high risk of banding and should be avoided.
Due to the non-linear encoding, blending may look significantly different. Strong colors may mix differently and a given opacity will create a different result.
Note that 32-bits allows negative values (which you might create by increasing saturation of a color) and 16-bits does not. While it is unlikely you'd ever see it, it is possible that some extreme HDR colors may show a slight color shift when the negative channels are clipped to 0. If you did run into it, just back off the saturation (such extreme colors can't be properly displayed anyhow).
Similarly, very bright HDR colors may be numerically different after round trip conversion to 16-bit HDR and then back to 32-bits. The difference is largely academic, as any difference should be visually minor or undetectable. However, if you use extreme values (those much brighter than any high end HDR monitor can handle), there may be some clipping since the 16-bit HDR does not encode the extreme ends of the HDR range.
Note: Photoshop does not render HDR correctly if PS was launched without an active HDR monitor (be careful when working with an HDR laptop connected to an external monitor). In this scenario, the 16-bit and 32-bit HDR may look substantially different, but it is a false preview based on PS not showing HDR in this scenario.