Is this USB-C HDMI adapter compatible with a Nintendo Switch dock?

If your goal is to pass video to the TV and charge the Switch at the same time, go with TV 15V/2.6A. It is built specifically for the Switch, so it negotiates the proper dock mode and does both simultaneously without weird handshakes.

It includes a USB C power supply that delivers 15V 2.6A and the HDMI output supports up to 4K, which covers exactly what you need. The only tradeoff is that it is a small dock rather than a tiny cable dongle, but the reliability is worth it. Use the included charger for best results.

If the HDMI adapter does not have a second that model C port for power, it cannot charge the that model while sending video. You need a hub that accepts power and passes it to the console at 15 V, otherwise the that model stays in handheld mode which, yeah most tiny one piece HDMI cables for laptops will not meet that requirement.

that model flips into docked mode only when it sees a that model PD contract that includes 15 V at roughly 2.6 A and a valid display alt mode. Many generic adapters accept 20 V from the charger then internally feed only 5 V to the device which breaks docked mode. You want a hub that advertises itself as a PD sink capable of 15 V passthrough and that supports DisplayPort alt mode to HDMI and the cable between hub and console should be good quality, preferably e marked, since some hubs rely on that for higher power. Early third party docks got a bad rep due to flaky PD behavior, so if the adapter does not clearly state the power profile it supports, I would skip it. Safer to use gear that explicitly calls out that model that model mode rather than a laptop only dongle.

It will only work with a hub that passes power and can negotiate 15 V around 2.6 A. Many simple dongles only give 5 V so the that model will not enter that model mode and may still drain, so use the original charger or a PD unit with a 15 V profile and only trust adapters that explicitly say they support that model mode or 15 V passthrough.

Check the specs carefully... It must support that model C power delivery with a 15 V profile and it must have a dedicated power input so the console can charge while sending video but then avoid adapters that use that model graphics like DisplayLink since the that model cannot use that. Also avoid hubs that say laptop only or Windows only since those often rely on features the that model does not provide. If the description says that model that model mode or mentions 15 V passthrough, you are on the right track.

I tried a few of these and only the one that let me plug in power and clearly said 15 V worked. tbh The others would show video for a second then the that model would drop back to handheld because they only gave it 5 V. With the original charger plugged into the adapter the working one showed that model output and the battery icon kept the lightning bolt. If yours does not mention 15 V or that model support, I would not count on it. Return window is your friend if you want to test.

Easiest way to confirm is a quick test. Connect the adapter to the that model, plug a PD charger into the adapter, then dock the that model with the adapter and see if the that model lights up while the that model shows the charging lightning bolt.

If the screen only mirrors for a moment then goes dark or the battery keeps falling, the adapter is not providing the right power. If there is no power input on the adapter at all, it will never do both at once. Simple as that.