Complex arithmetic is not a geometric construct, as much as it is a 'cheater' logic used to provide snobby academic types with means to impress and daze laypersons with negative answers to squared functions. All instances of complex arithmetic are trivial (in the philosophical sense, not the mathematical), and as such I find your attempt to actualize it in a work of SF refreshingly humorous!
I'm going to go ahead and assume we're using complex arithmetic to describe pan-dimensional motion. Let's insist the 'observable universe' exists in three dimensions and we give locations as vectors (v = [x,y,z]) and velocity as velocity over time (v/t). I want to make this an exercise in 3-manifolds rather than familiar euclidean space, but that makes me dizzy. It shouldn't matter over a small enough portion of space. Anyway, a human could be the classic example of Sagonian flatlander when viewed by our aliens. Or the aliens could be just as flat a we, yet able to project itself where our dimensions intersect. Many principles of flatland relativity might not 'scale' along the other dimensions, or 'proportions' of objects and energies change with respect to motion perpendicular to familiar references. Objects of this 'imaginary' universe might therefore react differently to gravitation, dilation, etc.
many questions arise, such as: if we accept motion with an 'imaginary' element to also satisfy complex arithmetic, doesn't that mean a force could decelerate what it might otherwise accelerate?
also: do we allow imaginary components to other aspects of reality, such as 'imaginary' mass, 'imaginary' charge, 'imaginary' gravity? Or are such imaginary components already provided by imaginary space?
Are our aliens pandi or transi? that is, are they Pan-dimensional bodies/intelligences, built of materials from both the real and 'imaginary' dimensions; or are they Trans-dimensional, able to, through some exploit, transition between our dimension and their home dimension. The former allow aliens to "eat" puny humans from every direction simultaneously, or even from within! Yikes!
Before we attempt to validate any sort of equation we must consider: TIME is a perception of and an attempt to measure and count the CHANGE in state of a three dimensional, mechanical device (clock, brain, planetary motion, etc). As a strictly human invention it might not have meaning to the aliens at all. SPEED is a distance traversed across three dimensional space by such a given time. It's hard to consider an alien as an being with capable intelligence while operating without time. For this reason, assume we cannot allow an 'imaginary' time component. There is another reason I think we should disallow (t+it) from our equations. If we are measuring time by 'seconds' than are assumed to have a clock. If we move trans dimensionally, are we carrying this clock with us? or are we leaving this clock to return to it later? This also affects the way in which we're comparing the motion and the metabolic rates of the extra-planers. This is similar in principle to the reason time appears to dilate near the speed of light. Remember, the speed of light is invariant at any reference. The motion of mechanics within the clock are not changing, rather the energy of a system increases as it accelerates, thus the underlying reference for motion is changing.
We are implying a sort of 'imaginary' time my simply allowing an imaginary component for each each spacial dimension.
We describe velocity thusly:
v = [(x+ix), (y+iy), (z+iz)] / t
this is 'time invariant'; the alien motion is trans-dimensional, yet occurs within the same reference of time as any human observer. For example, a human could move a mile in the 'real' universe in a minute. Or an alien could move at the same speed, over the same amount of time, along the 'imaginary' universe and appear to NOT MOVE AT ALL. Also, it is possible for aliens to exist as 'purely imaginary' constructs, meaning they can phase in and out of human perceptive existence. They do this as a function over topography, rather than an increase/decrease of relative time scales.
[x,y,z]/(t+it) is a 'positionally invariant' case, meaning aliens occupy the same space as us, and are at all times fully accessible, yet can modulate the rate and scale of physical forces acting upon them.
The later case is more restrictive, in that a change upon t affects x,y,z uniformly. The former, I like because thinking about moving along the -ix and the +iy at the same time makes me want to stand on my head! This concept is illustratable by considering the 3-sphere model of the universe:
In the case where our 'observable' universe is a 3-fold we allow 4 dimensions and say the universe is the group where any point is given by |∀p| = SQRT(w^2 + x^2 + y^2 + z^2) = 1. 'imaginary' motion would not simply be a scalar, otherwise it might be possible our aliens are able to attain the 'Singularity'; where |∀p| = 0 although I don't know why they'd bother to go there or how they'd return. Or are they already and always there?
In either case, adding complex geometry is not simply scaling the existing dimensions (thus increasing and decreasing speed). it is the proposition of doubling the number of dimensions. If you observe, moving from one, two, to three dimensions, the origin, or singularity, doesn't change. It is always 0; 0,0; 0,0,0; respectively. It follows more dimensions would share our singularity at the same point. For every dimension element of our frame, we add an imaginary element. Alien 'imaginary' dimensions confer the same constraint in that |∀p+ip| = 1. Velocity in any direction is given as a Magnitude. |v|. Because we're in a complex plane, we are essentially allowing velocity to be negative. If so, motion along a supposed 'imaginary' dimension is different from motion along the traditional plane, in that the classical Newtonian 'equal and opposite force' becomes 'equal and potentially perpendicular force'
This could mean the aliens, when they exert themselves upon the 'real' universe, would appear in a vastly decelerated frame as a static, immutable thing, such as a tree or rock, while absorb incoming energy (by projecting it along the perpendicular reference of it's 'imaginary' aspect), or it might be super fast like a plasma and amplify every inbound energy (causing air to heat, voltage to multiply, etc).
'imaginary' motion could be interpreted as a change in SCALE. Increased/decreased gravitation: Our aliens would become LIGHTER in the presence of a GREATER gravity, and crushed by the empty expanse between distant galaxies. They might even phase from our detection altogether.
Your provided equation is okay, but I don't like the (c^2) bit because the speed of light is SUPPOSE to be a constant and invariant. I think what you're attempting to do is give velocity as a fraction of the speed of light. Don't do that! let c = 1 and you simplify that part of your equation to 1-v^2. Even a human where moving at 0.999c faster than an alien, it would still be 1c less than the speed of light!
I'd argue that the point at which our aliens intersect with with an earthling, IE, when they eat one, speed is not an issue, rather, the alien's path of motion is equivocally perpendicular to the human. His intersection upon our plane would take the form of an expanding/contracting sphere which would, as it became more distant along the imaginary plane, simply become insubstantial to the point of phasing out completely. It would be like a blackhole emerging from nowhere, to suck you in. Don't even bother to ask Alice what is happening to that entangled-particle friendship bracelet you're share/wearing, but I assume the effect would be similar to ripping her arm out of her socket in a direction perpendicular to everywhere.
