Code by @harrisonturton: https://github.com/flutter/flutter/issues/17099#issuecomment-385870942

custom-bottom-navigation.dart

import 'package:flutter/material.dart';

class BottomNavigation extends StatefulWidget<_BottomNavigationState> {
	BottomNavigation({
		this.children,
		this.items,
		this.fadeDuration
	});
	final List<Widget> children;
	final List<Widget> items;
	final Duration fadeDuration;

	_BottomNavigationState createState() => new _BottomNavigationState();
}

class _BottomNavigationState extends State<BottomNavigation> with TickerProviderStateMixin {

	int _currentIndex = 0;
	int _prevIndex = 1;
	List<Widget> _items;

	@override
	void initState() {
		super.initState();
		_items = List.generate(widget.items.length, (i) {
			return new InkWell(
				onTap: () => _changePage(i),
				child: widget.items[i]
			);
		});
	}

	void _changePage(int index) {
		if (index == _currentIndex) return;
		setState(() {
			this._prevIndex = _currentIndex;
			this._currentIndex = index;
		});
	}

	List<Widget> _buildTransitions() => List.generate(
		widget.children.length,
		(int i) => new AnimatedOpacity(
			opacity: _currentIndex == i ? 1.0 : 0.0,
			duration: widget.fadeDuration == null ? new Duration(milliseconds: 150) : widget.fadeDuration,
			child: widget.children[i]
		)
	).toList();

	Widget build(BuildContext context) {
		return new Scaffold(
			body: new Stack(
				children: _buildTransitions()
			),
			bottomNavigationBar: new Row(
				mainAxisAlignment: MainAxisAlignment.spaceAround,
				children: _items
			)
		);
	}
}