An easy way to visualize a multiple regression with 2 independent variables is by a plane, as a plane is defined by 3 points that do not lie within the same line.
So we have the Y-intercept, the X variable (axis) and the Z variable (axis) to visualize three points making up a plane.
However a multiple regression is not limited to 2 independent variables. How do you visualize a multiple regression given, say, 3 independent variables?
It's possible to visualize a multiple regression with 2 predictors without using a 3D plot (which is implied by your discussion of the plane & Z-axis). Instead of the Z axis, we typically use colour to indicate variation in the extra dimension. The result is called a level plot (or a contour plot if contours are used instead of colour). Here's an example I found via google:
These plots are very useful but the term 'level plot' is not very well-known. It's possible that this goes by other names as well, but I'm not aware of them.
This plotting approach generalizes easily to 3 dimensions. Imagine that there was a third predictor in the model: Sex. In that case, just use one level plot for each Sex. Or imagine that the predictor is continuous, such as 'Daily Calorie Consumption' or Consumption for short. In that case, choose a small number of levels of Consumption (say 5) that meaningfully cover the range of values you are interested in. Use one level plot to convey the model-predicted Age x Weight interaction at each of these Consumption levels.
I have used this for 4 dimensions too, leading to a nice plot matrix. Beyond 4, I think its usefulness starts to break down. But 5-dimensional visualization is a major challenge, even though it's possible to use time (animation) as one dimension.
