Indeed the answer is that you do land a little bit away from where you start, but it’s such a small amount that the error in how vertical you jump is larger than the amount you would shift due to the rotation of the earth.

Remember, the rotation of the earth is pretty fast, but you are going the same speed as it. When you jump up, you still have that momentum taking you along. The point though is that as you jump up, even if there is no air resistance, because you are further from the centre of the earth you need to be moving a bit faster than the surface of the earth to keep the same rotational speed. Imagine two races going around a circular track. The one furthest from the centre needs to be going faster than the one closer in for them to remain side by side. So in this example the surface of the earth will be rotating slightly faster than you – that is, at the same horizontal speed the earth will turn a circle slightly quicker than you will when you jump off the surface. However, the difference is roughly 1/1000000 in your angular rotation speeds, so they are basically the same and so in the short time that you are up and down you will land back in more or less the same place.

If you tried to shoot a rocket directly up a km or so, you would indeed notice the difference and it would not fall back directly where it started, but again in this case air movement and being slightly off vertical will have quite a big effect, so you need to go up a long way to really see the effect strongly.