Would my helicopter land in a different place?

First, a helicopter cannot fly to a level where not bothered by gravity.
Second, the atmosphere moves with us.
This doesn’t require a great knowledge of physics.

Okay, i gigured that much.
What i forgot to put in my question was: imagine that my copter could do just that.

It depends on how you define “straight up vertical.”

If you stay above the same point on the Earth for 12 hrs, then of course you would land in the same place.

If you define straight up in relation to some fixed point in relation to the stars, then the Earth would turn below you and you would land in a different place.

I think that there would at the very least be a minute shift to compensate for advance in orbit.
Interesting question. I used to wonder about time travel and how one would have to be very careful where in time one projected oneself as it could be lethal to materialize into a concrete wall.

We can think of this much easier if you think of the Earth as a spinning plate and you as a weight clipped to the surface. Now suppose there is a string (not attached to the surface of the plate but to the axis of rotation of the plate).

You spin the plate up to a certain speed. This is analogous to the rotation of the Earth. With the weight attached to the edge of the plate you are traveling at the same speed as the surface.

Now the clip releases the weight. The string will prevent the weight from flying away but will constrain it to rotate around the center of the plate.

The rotation of the weight on the string will be slower than the rotation of the plate. If you were to reel in the string the weight would “land” behind the location it left.

The reason this happens is because rotational inertia is conserved. Rotational inertia is a function of weight, linear velocity, and distance from the center of rotation. Since the mass of the weight doesn’t change, when the distance to the center of rotation was changed the velocity had to reduce to conserve rotational inertia.

tldr: Yes you would land in a different place. You and the Earth will spin at differing rates the higher you go.

Once aloft in an aircraft, your natural frame of reference is the air surrounding you.

A stunt you can readily pull off with a Cessna: go up on a very windy day, fly into the wind and throttle back to just above stall speed. Negative ground speed!

As the universe continues to expand at about 1,000,000 miles each day you would likely land on one of the larger moons of Jupiter. I’m guessing Ganymede.

@AstroChuck Humorous answer, but in reality you cannot determine a fixed point of the universe itself.