"Field has incomplete type" error

You are using a forward declaration for the type MainWindowClass. That's fine, but it also means that you can only declare a pointer or reference to that type. Otherwise the compiler has no idea how to allocate the parent object as it doesn't know the size of the forward declared type (or if it actually has a parameterless constructor, etc.)

So, you either want:

// forward declaration, details unknown
class A;

class B {
  A *a;  // pointer to A, ok
};

Or, if you can't use a pointer or reference....

// declaration of A
#include "A.h"

class B {
  A a;  // ok, declaration of A is known
};

At some point, the compiler needs to know the details of A.

If you are only storing a pointer to A then it doesn't need those details when you declare B. It needs them at some point (whenever you actually dereference the pointer to A), which will likely be in the implementation file, where you will need to include the header which contains the declaration of the class A.

// B.h
// header file

// forward declaration, details unknown
class A;

class B {
public: 
    void foo();
private:
  A *a;  // pointer to A, ok
};

// B.cpp
// implementation file

#include "B.h"
#include "A.h"  // declaration of A

B::foo() {
    // here we need to know the declaration of A
    a->whatever();
}

The problem is that your ui property uses a forward declaration of class Ui::MainWindowClass, hence the "incomplete type" error.

Including the header file in which this class is declared will fix the problem.

EDIT

Based on your comment, the following code:

namespace Ui
{
    class MainWindowClass;
}

does NOT declare a class. It's a forward declaration, meaning that the class will exist at some point, at link time.
Basically, it just tells the compiler that the type will exist, and that it shouldn't warn about it.

But the class has to be defined somewhere.

Note this can only work if you have a pointer to such a type.
You can't have a statically allocated instance of an incomplete type.

So either you actually want an incomplete type, and then you should declare your ui member as a pointer:

namespace Ui
{
    // Forward declaration - Class will have to exist at link time
    class MainWindowClass;
}

class MainWindow : public QMainWindow
{
    private:

        // Member needs to be a pointer, as it's an incomplete type
        Ui::MainWindowClass * ui;
};

Or you want a statically allocated instance of Ui::MainWindowClass, and then it needs to be declared. You can do it in another header file (usually, there's one header file per class).
But simply changing the code to:

namespace Ui
{
    // Real class declaration - May/Should be in a specific header file
    class MainWindowClass
    {};
}


class MainWindow : public QMainWindow
{
    private:

        // Member can be statically allocated, as the type is complete
        Ui::MainWindowClass ui;
};

will also work.

Note the difference between the two declarations. First uses a forward declaration, while the second one actually declares the class (here with no properties nor methods).