One tool I find very useful for creating movement patterns is the timeline asset. The advantage of using timelines for this is that you can describe the movement patterns using curves in a visual editor instead of describing them as mathematical functions or algorithms. This is usually far more intuitive and designer-friendly.
You can even do it in the Unity editor. When you want the boss to be in the upper left corner at second 2 and in the center of the screen at second 3, then you:
- enable recording for that object in the timeline editor
- move the timeline to second 2
- move the object to the first position using the usual editor controls,
- move the timeline to second 3,
- move the object to the second position.
The timeline editor will now have generated curves for the x and y coordinate which smoothly interpolate between these positions. You can then tweak these movement curves by adding additional points in between or moving points around.
For example, my entry "Slacking At Work Simulator" for the Useless Simulation Game Jam two weeks ago on itch.io makes a lot of use of timelines. The most visible example is the movement of the antagonist. His x position, y position and rotation is controlled completely by one 5 minute long timeline.
However, timelines have one crucial disadvantage: They can not react to what happens in the game.
Going back to the boss monster from the youtube video in the question: It appears to me as if the y-position of the opponent follows a fixed pattern while the x-position reacts to the player position. So you could combine a timeline which controls transform.y with a script which controls transform.x by always moving towards the current position of the player.