Are all images 2d or also we have 3d images?
The term image derive from  Latin  imago. I fully agree with wikipedia saying that it  denotes an artifact


  that depicts visual perception [...] that resembles a subject—usually
  a physical object—and thus provides a depiction of it.


As such, it is a representation of some reality, combining spatial "ordinal" cooordinates (with some natural order, in 2D and 3D for us humans) and  "cardinal" coordinates (luminance/chrominance values, spectra, categories).

The dimension attributed to images to signal/processing people is generally the one of the "ordinal" cooordinates (2D or 3D). To other practitioners who plot 2D images as 3D plots (using elevation for scalar fields in the "cardinal" dimension, non-image-processing people often call 2D image "tridimensional" (bottom-right "Arnold Schwarzenegger" in Terminator), because they look at this artifact in our natural 3D world:

.

Here, $x$ and $y$ are the ordinal dimensions, $f(x,y)$ makes the third. This visualization of images in the resulting topological/topographic map is exploited for instance in the field called "mathematical morphology",  where surfaces and regions are interpreted as geographical objects: valleys, mountains, watersheds, erosions, etc.

If you add (a 1D) time (a very ordinal dimension so far) for motion, then you can add another dimension: standard 2D+t video, 3D video, etc. 

If we call for scale-space theory, where one bluntly considers that "objects are meaningful over a certain range of scales only", one can smooth a given image with smoothing kernels to get different images: 



and pile them up to yield an additional "scale" dimension. Then, you can extend the discussion to images at arbitrary coordinates, like point clouds, with motion, etc.

Finally, I believe that image data (and their dimensions) can be considered as at least three objects:


how they should be in reality (if such a thing exists),
how they are modeled or described,
how we process them.


and their inherent dimension depends on that. Remember however that one can map two continuous spaces of different dimensions, albeit in a non continuous manner. For discrete spaces, like the ones for classical image/volume coordinates, mappings exist, like zigzag of Hilbert/Peano curves for image compression (JPEG, J2K), so dimensions are a matter of data structuration.