Trying to work out how to beat the countermeasure level, the two beam artillery guns always shoot me when I get near, no matter where i am in the 4th dimension.

I tried sneaking up on them with the shotgun whilst far away in the 4th blue dimension, I see the beam being fired and killing me, even when the blue markers for the two guns are near the bottom of the circles!

Is this a bug or do the beam guns cheat intentionally?

That's not their rotation relative to you, that's your rotation relative to them. The only way to judge which way they're pointing is to look at them. Are you sure you're moving in 4D, or just rotating?

Yep, just because you're not aiming at them doesn't mean they're not aiming at you . As a quick hint, maybe take another look at the description text at the start of the mission.

So when you aim in the hidden axe, you are actually not moving in that axe? What I mean is, can I shoot to z = 10 from z = 0 ? Man I have to think quite a lot about it to understant rotation and movement in the hidden axe...

How do I find out where am I respect the beam artillery? That is, how can I dodge the beam artillery using the hidden axis?

Right. If you're facing an enemy and he fires at you, you're simply turning to look away from the bullet, not dodging it.

You shoot where you're pointing, so if you're pointing towards positive Z, your shot will hit Z ₁₀ . If you're pointing towards negative Z, your shot will instead travel to Z _-10 .

You can't tell unless you're looking at the artillery, because there are no landmarks in space. If you move along X it shifts away from the center of your vision along X. In order to keep it in the center, you'd have to rotate the opposite way through X. The same thing applies to the other axes besides W, the depth axis. So if you start out with the enemy aligned on all three dots and then that ceases to be the case, either:

You are rotating
You are moving
The target is moving

See also my post in the 4D Fundamentals topic on strafing:
http://www.mushware.com/viewtopic.php?p=465#465

Really, you think about common sense and past experience with how movement and rotation works in 3D, and test to see if the same applies in 4D. In the first level, perform various motions in the visible axes and watch the effect on the dot markers of the drones. Then try and mimic the effect with the hidden axis, or if you're using two axes, one visible and one hidden.