I've managed to visualize a Klein bottle in 4d. I easily visualize 3d objects. However I can't really do color - I startled myself recently when I briefly saw red. On that aphantasia test with an apple, I can hold it's 3d shape, but no surface texture or color.
People seem to have surprisingly different internal experiences. I don't know how common 4d visualization is, and I suspect even those capable require exposure to the concepts and practice. However I do think it possible.
Your hippocampus has several special clusters of neurons whose members activate and deactivate based on your body's understanding of your position and momentum in a 3D world.
The arrangement of these neurons physically corresponds to reality, and so things are pretty hardwired.
Repurposing these neurons might be possible with advanced training and nootropics, but I'm not sure. You might have better luck engaging other parts of your brain, for example using metaphor or abstraction such as mathematics.
For me, being able to visualize 4D would imply that I can picture four mutually perpendicular axes, something which I find completely impossible for me to do. And I thought it is impossible for any human brain. It would be fascinating if I am wrong.
The blind French mathematician Bernard Morin is well-known for creating the first visualization of a sphere eversion, a method for turning a sphere inside out without creasing it. His work was based on Stephen Smale's 1958 proof of sphere eversion's existence and on ideas shared by Arnold Shapiro. Morin's method involved constructing a sequence of models, including his "Morin surface," to demonstrate the process.
https://en.wikipedia.org/wiki/Bernard_Morin