How can I work with dynamically-allocated arbitrary-dimensional arrays?

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There is no need to reinvent the wheel, C has that since C99, it is called variable length array, VLA. It has just the syntax as “normal” d-dimensional arrays, only that the bounds may be variable and they are not allowed in file scope.

As such objects can become relatively large, you should not allocate them on the stack, but with something like malloc

double (*A)[n][m] = malloc(sizeof(double[k][n][m]));

The compiler then helps you with all the index calculations without problems. If you want to pass such animals to functions you just have to be careful to declare the bounds first:

void func(size_t k, size_t n, size_t m, double A[k][n][m]);

This makes your intent clear to both, the human reader and the compiler. I prefer this over the equivalent form

void func(size_t k, size_t n, size_t m, double (*A)[n][m]);

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