Recommendations for calendar with iOS or Android sync and ignores time zones

I am commuting between the West Coast and the East Coast about once a week. The time zones are 3 hours apart. I enter most of my appointments on a PC with Google calendar, some on the iPhone and some on an Android tablet. The iPhone switches time zones automatically, the other ones don't so I have to switch manually (sometimes I do, sometimes I forget). I often make appointments for one coast while I reside on the other coast and when I agree to meet somebody at 10am this means the local time of wherever the meeting will be, not the time zone adjusted time.

So it happens all the time that when I enter an appointment it gets displayed by three hours off on the appointment date. My calendar is one big mess by now and I have zero confidence for it to show me the right time for an appointment.

What I would like to see is a mode where the calendar shows whatever time I have entered it at no matter the current time zone. Interestingly, paper calendars do this extremely well.

I would think other travelers have the same problem. Flight departure and arrival times always are quoted in local time so you can't just enter them into your calendar without setting time zones.

I hope this makes sense and maybe somebody has recommendations for either different calendar software or a different system. Please keep in mind that I just want the calendar to display my times the way I entered them without changes. Having to set time zone for each appointment or different calendars for different time zones wouldn't be a good solution because it would add more complexity and is also error prone.