To find the zone or to be in the flow, the ideal task to choose would be something that lies in the sweet spot between boredom and anxiety. Choose something slightly more challenging than your current skill level.
You'll feel bored if your current skills far outweigh what you have chosen to work on. You'll feel anxious and give up if you feel it's way too challenging and your skill level is much lower than required.
e.g., if you want to become a chess champion, practising your moves against a primary school student won't get you in the flow. Practising against Bobby Fischer may scare you off chess forever. Find someone at a local chess club at a slightly higher level than you and you're likely to get into the flow.
Switching among different screens per se is not contradictory to deep work. If you are working on an absorbing research project, you may need to refer multiple sources and that's OK. What you need to check though is whether this task that requires you to switch screens is really a deep work task, or shallow work.