I think color of the image would be considered a 3rd dimension, because you'd have to represent the pixels by their x, y, and color coordinates.  In fact, pixels/voxels are usually represented as higher dimensional vectors depending on how many features they have.

But the dimension is not 3D in space, it's 3D in a more abstract sense.

Imagine to observe a plane with a "2D camera" (this avoid an observer who is external to the system - whom considerations might be affected by the perception of more dimensions). A sufficient and necessary condition to determine whether or not an object belongs to that plane is, while the object is moving, if there is always a possible movement of the camera such that it is kept centered in the lens. To help visualize this, imagine a crocodile peeking at water level its prey, following it preserving a certain distance in order not to alert it. Outside, you would se two entities moving while maintaining the same [diametrical] distance between them. 

In the case you propose, then, is such a vectorial movement possible? If the prey dives, the vector's direction connecting them would never belong to the plane: so, within a 2D space, you prove that it does not [exclusively] belong to the plane! On the other hand, an outside observer would comment it appears as a projection of an object belonging to a system of reference with [at least] another dimension. 

In the end, this may be a evidence for something holographic - furthermore, something like this appearing and disappearing, it is observed in the context of elementary particles... 

Federico Bugnoli, thank you for the answer. Are you saying that if something is 2D or 3D depends from the observer: if he or she is able to see another dimension?

Adam Hughes, thank you for your answer, does that imply that we can not have moving object in  just 2D? What is the lowest level of 3D, when exactly is 3D becoming 2D?

As Adam Hughes said, 3D "in geneal" means you have three features that univocally describe a object. If you assign two spatial coordinates and the feature "color" to a spot, you could justify your scenario through a switch between "black" and "transparent"; nevertheless, your dot would formally be a 3D object, as for I just said. 

Let's forget about it being colored: simply consider it "being measurable" and it "being not measurable". 

Within a 2D perspective, depth effect is rendered in 1D (remember the previous example of the crocodile): the "2D camera" shoots a 1D-segment becoming wider or narrower as the "black spot" approaches or moves away - the same happens in 3D, but the effect is [intuitively] rendered in 2D (here's a example, http://cdn-1.vivalascuola.it/o/j/come-disegnare-un-parallelepipedo-in-prospettiva-centrale_471368a77ee0685d37c5633fc27b00f4.jpg consider the faces parallel to the screen: as planes, they differ by the value of two coordinates, while their "thickness" is the same, punctiform). Both in 2D and 3D you could explain the "appearance and disappearance" of a object as its going very quickly from its current position to an infinite coordinate, then going back to where it was; but, you know, speaking of "going to infinite" turns up one's nose, even if the laws of perspective justify what the "2D camera" would shoot.

To answer your question, if you are a 3D observer, it's extremely probable you would assist to "the disappearing of the black dot" outside the plane in which it exists (you watch it from above or below), so that this plane is actually inscribed in a 3D world, and consequently the dot has three spatial coordinates - this for justifying my "2D camera" approach, in order to avoid "third axis diversion": in 3D, you can justify your scenario in many ways (i.e. its crossing some planes back and forth), but if you want to describe it as happening in 2D, you must look at it in the place of a 2D observer. Something may have 4[spatial]D properties even if you are a 3D observer, but what you call it, it is a 3D object, since you cannot perceive the fourth axis and its effects; likewise, it may be 2D, but you would see it as 3D - to describe its 2D properties, you must forget the third. As if it was simple... 

EDIT(ADD)//< Keeping in mind what I previously wrote, even if the dot goes to infinite, we may allow the camera to follow it, being them both within the plane. That said, the "diametrical proof" I provided continues to apply: there is no value such that it is the measurable result of infinite minus infinite (the way we measure the distance between). On the other hand, we may describe any point in the plane with (x;y;0), such that the last coordinate isn't relevant for the identification of any spot (it would be formally given by three features, but the last is homogeneous, so it can be considered undefinitive and omitted), then a dot to infinite is given by (1;1;1) - I'm assuming every value to the infinite converges into a constant: this means the camera wouldn't ever be able to keep the distance d>0 with respect to the black spot it had throughout the plane. 

I just randomly realized something. By definition, a "point" is merely a vector of coordinates: you can consider a 2D or a 3D universe, but it is anyway nothing but a "position". So, let me discuss the "disappearence of a dot" instead of "a black dot". 

Following the previous argumentation, the key is whether an abserver can measure it or not. But why wouldn't I be able to? It depends on what we mean by "space", and in particular its cardinality: discrete mathematics works with natural numbers, physics at least with reals, and these two sets have different cardinality, and |N^2|