For my code I am attempting to get an array of AXMenuItems from an AXMenu (AXUIElementRef). The menu logs successfully, and here is my code:
NSArray *anArray = [NSRunningApplication runningApplicationsWithBundleIdentifier:@"com.apple.dock"];
AXUIElementRef anAXDockApp = AXUIElementCreateApplication([[anArray objectAtIndex:0] processIdentifier]);
CFTypeRef aChildren;
AXUIElementCopyAttributeValue(anAXDockApp, kAXChildrenAttribute, &aChildren);
SafeCFRelease(anAXDockApp);
CFTypeRef aMenu = CFArrayGetValueAtIndex(aChildren, 0);
NSLog(@"aMenu: %@", aMenu);
                    // Get menu items
CFTypeRef aMenuChildren;
AXUIElementCopyAttributeValue(aMenu, kAXVisibleChildrenAttribute, &aMenuChildren);
for (NSInteger i = 0; i < CFArrayGetCount(aMenuChildren); i++) {
    AXUIElementRef aMenuItem = [self copyAXUIElementFrom:aMenu role:kAXMenuItemRole atIndex:i];
    NSLog(@"aMenuItem: %@", aMenuItem); // logs (null)
    CFTypeRef aTitle;
    AXUIElementCopyAttributeValue(aMenuItem, kAXTitleAttribute, &aTitle);
    if ([(__bridge NSString *)aTitle isEqualToString:@"New Window"] || [(__bridge NSString *)aTitle isEqualToString:@"New Finder Window"]) /* Crashes here (i can see why)*/{
        AXUIElementPerformAction(aMenuItem, kAXPressAction);
        [NSThread sleepForTimeInterval:1];
        break;
    }
}
What is the correct way to get the list of AXMenuItems? 
Screenshot of the Accessibility Inspector:

I have figured out an answer, using @Willeke answer of using AXUIElementCopyElementAtPosition() to get the menu. Since there were multiple dock orientations and hiding, I had to create enums in the .h file as it would be easier to read than 0, 1, or 2.
// .h
typedef enum {
    kDockPositionBottom,
    kDockPositionLeft,
    kDockPositionRight,
    kDockPositionUnknown
} DockPosition;
typedef enum {
    kDockAutohideOn,
    kDockAutohideOff
} DockAutoHideState;
Then, I added the methods to get these states in the .m
// .m
- (DockPosition)dockPosition
{
    NSRect screenRect = [[NSScreen mainScreen] frame];
    NSRect visibleRect = [[NSScreen mainScreen] visibleFrame];
    // Dont need to remove menubar height
    visibleRect.origin.y = 0;
    if (visibleRect.origin.x > screenRect.origin.x) {
        return kDockPositionLeft;
    } else if (visibleRect.size.width < screenRect.size.width) {
        return kDockPositionRight;
    } else if (visibleRect.size.height < screenRect.size.height) {
        return kDockPositionBottom;
    }
    return kDockPositionUnknown;
}
- (DockAutoHideState)dockHidden
{
    NSString *plistPath = [NSHomeDirectory() stringByAppendingPathComponent:@"Library/Preferences/com.apple.dock.plist"];
    NSDictionary *dockDict = [NSDictionary dictionaryWithContentsOfFile:plistPath];
    CFBooleanRef autohide = CFDictionaryGetValue((__bridge CFDictionaryRef)dockDict, @"autohide");
    if (CFBooleanGetValue(autohide) == true) {
        return kDockAutohideOn;
    }
    return kDockAutohideOff;
}
For the dock position unknown, I added in case it was not able to calculate it from the screen positions.
Then, I used a method to get the dock item from the menubar:
- (AXUIElementRef)getDockItemWithName:(NSString *)name
{
    NSArray *anArray = [NSRunningApplication runningApplicationsWithBundleIdentifier:@"com.apple.dock"];
    AXUIElementRef anAXDockApp = AXUIElementCreateApplication([[anArray objectAtIndex:0] processIdentifier]);
    AXUIElementRef aList = [self copyAXUIElementFrom:anAXDockApp role:kAXListRole atIndex:0];
    CFTypeRef aChildren;
    AXUIElementCopyAttributeValue(aList, kAXChildrenAttribute, &aChildren);
    NSInteger itemIndex = -1;
    for (NSInteger i = 0; i < CFArrayGetCount(aChildren); i++) {
        AXUIElementRef anElement = CFArrayGetValueAtIndex(aChildren, i);
        CFTypeRef aResult;
        AXUIElementCopyAttributeValue(anElement, kAXTitleAttribute, &aResult);
        if ([(__bridge NSString *)aResult isEqualToString:name]) {
            itemIndex = i;
        }
    }
    SafeCFRelease(aChildren);
    if (itemIndex == -1) return nil;
    // We have index now do something with it
    AXUIElementRef aReturnItem = [self copyAXUIElementFrom:aList role:kAXDockItemRole atIndex:itemIndex];
    SafeCFRelease(aList);
    return  aReturnItem;
}
This SafeCFRelease() method is a very simple method that checks if the passed value is not nil, then releases (had some issues earlier).
void SafeCFRelease( CFTypeRef cf )
{
    if (cf) CFRelease(cf);
}
And this method [copyAXUIElementFrom: role: atIndex:] is a method from @Willeke answer from another one of my questions:
- (AXUIElementRef)copyAXUIElementFrom:(AXUIElementRef)theContainer role:(CFStringRef)theRole atIndex:(NSInteger)theIndex {
    AXUIElementRef aResultElement = NULL;
    CFTypeRef aChildren;
    AXError anAXError = AXUIElementCopyAttributeValue(theContainer, kAXChildrenAttribute, &aChildren);
    if (anAXError == kAXErrorSuccess) {
        NSUInteger anIndex = -1;
        for (id anElement in (__bridge NSArray *)aChildren) {
            if (theRole) {
                CFTypeRef aRole;
                anAXError = AXUIElementCopyAttributeValue((__bridge AXUIElementRef)anElement, kAXRoleAttribute, &aRole);
                if (anAXError == kAXErrorSuccess) {
                    if (CFStringCompare(aRole, theRole, 0) == kCFCompareEqualTo)
                        anIndex++;
                    SafeCFRelease(aRole);
                }
            }
            else
                anIndex++;
            if (anIndex == theIndex) {
                aResultElement = (AXUIElementRef)CFRetain((__bridge CFTypeRef)(anElement));
                break;
            }
        }
        SafeCFRelease(aChildren);
    }
    return aResultElement;
}
Taking all this code, I put it into one of my methods:
// Check if in dock (otherwise cant do it)
                if ([self isAppOfNameInDock:[appDict objectForKey:@"AppName"]]) {
                    // Get dock item
                    AXUIElementRef aDockItem = [self getDockItemWithName:[appDict objectForKey:@"AppName"]];
                    AXUIElementPerformAction(aDockItem, kAXShowMenuAction);
                    [NSThread sleepForTimeInterval:0.5];
                    CGRect aRect;
                    CFTypeRef aPosition;
                    AXUIElementCopyAttributeValue(aDockItem, kAXPositionAttribute, &aPosition);
                    AXValueGetValue(aPosition, kAXValueCGPointType, &aRect.origin);
                    SafeCFRelease(aPosition);
                    CFTypeRef aSize;
                    AXUIElementCopyAttributeValue(aDockItem, kAXSizeAttribute, &aSize);
                    AXValueGetValue(aSize, kAXValueCGSizeType, &aRect.size);
                    SafeCFRelease(aSize);
                    SafeCFRelease(aDockItem);
                    CGPoint aMenuPoint;
                    if ([self dockHidden] == kDockAutohideOff) {
                        switch ([self dockPosition]) {
                            case kDockPositionRight:
                                aMenuPoint = CGPointMake(aRect.origin.x - 18, aRect.origin.y + (aRect.size.height / 2));
                                break;
                            case kDockPositionLeft:
                                aMenuPoint = CGPointMake(aRect.origin.x + aRect.size.width + 18, aRect.origin.y + (aRect.size.height / 2));
                                break;
                            case kDockPositionBottom:
                                aMenuPoint = CGPointMake(aRect.origin.x + (aRect.size.width / 2), aRect.origin.y - 18);
                                break;
                            case kDockPositionUnknown:
                                aMenuPoint = CGPointMake(0, 0);
                                break;
                        }
                    } else {
                        NSRect screenFrame = [[NSScreen mainScreen] frame];
                        switch ([self dockPosition]) {
                            case kDockPositionRight:
                                aMenuPoint = CGPointMake(screenFrame.size.width - 18, aRect.origin.y + (aRect.size.height / 2));
                                break;
                            case kDockPositionLeft:
                                aMenuPoint = CGPointMake(screenFrame.origin.x + 18, aRect.origin.y + (aRect.size.height / 2));
                                break;
                            case kDockPositionBottom:
                                aMenuPoint = CGPointMake(aRect.origin.x + (aRect.size.width / 2), screenFrame.size.height - 18);
                                break;
                            case kDockPositionUnknown:
                                aMenuPoint = CGPointMake(0, 0);
                                break;
                        }
                    }
                    if ((aMenuPoint.x != 0) && (aMenuPoint.y != 0)) {
                        AXUIElementRef _systemWideElement = AXUIElementCreateSystemWide();
                        AXUIElementRef aMenu;
                        AXUIElementCopyElementAtPosition(_systemWideElement, aMenuPoint.x, aMenuPoint.y, &aMenu);
                        SafeCFRelease(_systemWideElement);
                        // Get menu items
                        CFTypeRef aMenuChildren;
                        AXUIElementCopyAttributeValue(aMenu, kAXVisibleChildrenAttribute, &aMenuChildren);
                        NSRunningApplication *app = [[NSRunningApplication runningApplicationsWithBundleIdentifier:[appDict objectForKey:@"BundleID"]] objectAtIndex:0];
                        for (NSInteger i = 0; i < CFArrayGetCount(aMenuChildren); i++) {
                            AXUIElementRef aMenuItem = [self copyAXUIElementFrom:aMenu role:kAXMenuItemRole atIndex:i];
                            CFTypeRef aTitle;
                            AXUIElementCopyAttributeValue(aMenuItem, kAXTitleAttribute, &aTitle);
                            // Supports chrome, safari, and finder
                            if ([(__bridge NSString *)aTitle isEqualToString:@"New Window"] || [(__bridge NSString *)aTitle isEqualToString:@"New Finder Window"]) {
                                AXUIElementPerformAction(aMenuItem, kAXPressAction);
                                NSInteger numberOfWindows = [self numberOfWindowsOpenFromApplicationWithPID:[app processIdentifier]];
                                // Wait until open
                                while ([self numberOfWindowsOpenFromApplicationWithPID:[app processIdentifier]] <= numberOfWindows) {
                                }
                                break;
                            }
                        }
                        SafeCFRelease(aMenu);
                        SafeCFRelease(aMenuChildren);
                    }
                }
This is pretty complicated, but it works. I probably can't explain it, but I have stress tested this code and it works quite well.
Answer to the question "How to get an array of AXMenuItems from AXMenu?": aMenuChildren is the list of menu items. To be sure you could filter by role kAXMenuItemRole.
Answer to the question "Why does it not work?": The menu isn't a menu. When you inspect the menu in Axccessibility Inspector, it displays a warning "Parent does not report element as one of its children". The Dock app has one child, a list of dock items.
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