Long story short, you'll get great video just by opening the camera app and hitting record. You have several interesting options though if you chose to dig into the settings a little more. If you switch from Intelligent Auto to Manual shooting mode you gain the option to set resolution, frame rate, HDR on/off, and some other controls like white balance.
There is another mode which lets you record in 4k. This is somewhat limited though as the phone starts to heat up quite quickly. I've never had it shut down on me, but others have mentioned that you can only record about 10 minutes before things start to get too hot. Something else I noticed is that the field of view changes based on resolution.
First up is the Intelligent Auto mode. This records 1080p30 with no option to change the resolution or frame rate. While this mode is quite good, one thing that bothered me was how quickly it was willing to refocus. It doesn't say what the focus mode is in Auto but I'm going to guess it's some sort of face detect. It also has SteadyShot turned to Intelligent active by default. I think this mix of settings works well for most situations.
Intelligent Auto on a small tripod during the daytime.
Intelligent Auto handheld at night.
Next is the 1080p60 option in the Manual mode. For the videos below I left most of the settings to their default. In the future I will probably use Single Autofocus for anything I shoot in Manual mode. The 60 fps option works quite well. I noticed no loss of motion resolution, video artifacts, or any other anomalies that would prevent me from using this mode. Now that YouTube offers a 60p option it makes it even more tempting.
1080p60 on a small tripod during the daytime.
1080p60 handheld at night.
1080p60 pressed against the glass on the subway in Seoul.
Lastly is the 4k mode. This is 30fps and not that different from the auto mode. The first thing I noticed about this mode was that the field of view changed. It would probably take a lot of processing power to record at native resolution and down convert, so this makes sense, I've just never noticed it before in other phones. I don't have a 4k display, but my monitor is 1440p and the difference in resolution is noticeable in the raw files. After YouTube compression the difference is less noticeable. There also doesn't seem to be any difference in color space so I think the only thing that really changes from this and iA is resolution. Otherwise it looks fine. I like that the option is there, but I doubt I will use it very much. It might be a good way to show off that new 4k tv of yours though.
4k on a small tripod during the daytime.
4k handheld at night.
I hope this gives you an idea of how well the Sony Xperia Z3 Compact takes video. It has minimal quirks and works quite well in any mode.