I don't know if that's what you're looking for, but here are some equivalent ways to visualize spheres. I added some sketches, I hope they help somewhat.
One standard way to think of an n-dimensional sphere is by taking an n-dimensional ball and identifying all the points of its boundray. For example, for $S^2$, we have a $2$-dimensional disk, which has $S^1$ as a boundary, and in order to get the sphere we glue the boundary together to a single point (Top two drawings in the first sketch).
Performing the same for $S^3$, we take a $3$-dimensional disk (a ball) and glue all its boundary ($S^2$) to a single point. Those two examples are a special case of what is called a CW-decomposition for a manifold.
Note that the above construction is equivalent to the following: Take an open $n$-dimensional disk, and consider it as $\mathbb{R}^n$, to which it is homeomorphic. Define "a point at infinity" as the limit of all sequences of points on the disk the distance of which from
the center grows to infinity.
For example, in the case of $S^1$, we can take an open interval (identified with $\mathbb{R}$) and define a point $\infty$ as the limit of unbounded sequences.
In the case of $S^2$, and using the fact that the open $2$-dimensional disk is homeomorphic to $\mathbb{R}^2$, we get a gluing of a point "at infinity" to $\mathbb{R}^2$. The same construction performed with $\mathbb{R}^3$ results in $S^3$.
This representation is a common and useful one (illustrated in the bottom two drawings, and the one on the right, in the first sketch).
Another useful and intuitive ways to think of the $3$-sphere (and in general, compact oriented 3-dimensional manifolds) is looking at its decomposition into two $3$-manifolds which are simpler or easier to visualize than the original manifold. As you mentioned, one such example is decomposing $S^3$ into two $3$-balls glued along their boundaries (both of which are $2$-spheres).
While in the example of decomposing $S^2$ into two disks glued along their boundary ($S^1$) the visualization is easy, the case of $S^3$ might not be as intuitive, in which case let me suggest the following intuition: As pictured, take two balls, and while leaving one as it is, cut out a small ball in the center of the second one (in red). Then as we glue the boundaries together, the blue ball is "inside", while in the black one, the closer a point is to the center, the closer it is to infinity in the glued version (that's why it's denoted by $\infty$ in the sketch). As the radius of the red ball goes to zero, the radius of the resulting glued open ball goes to infinity, giving the previously described visualization.
A more complicated, albeit intuitive in my opinion, visualization is by taking two full genus-$1$ tori, and gluing them along the boundaries (which are regular genus-$1$ tori), gluing meridians of one torus to longitudes of the other. This is similar to the construction with the balls in the previous example. The following sketch demonstrates how the longitudes of the full tori look in such a gluing. Note that again, each such longitude has to "close" at some point, where the (red) one in the center closes at $\infty$.
The last two constructions are special cases of what is called Heegaard decompositions, or splittings.
