Following this guide https://git.coolaj86.com/coolaj86/ssl-root-cas.js/src/branch/master/Painless-Self-Signed-Certificates-in-node.js.md, I've created a Root CA and Signed Certificate with the following script:
make-certs.sh
#!/bin/bash
FQDN=`hostname`
# make directories to work from
rm -rf certs
mkdir -p certs/{server,client,ca,tmp}
# Create your very own Root Certificate Authority
openssl genrsa \
  -out certs/ca/my-root-ca.key.pem \
  2048
# Self-sign your Root Certificate Authority
# Since this is private, the details can be as bogus as you like
openssl req \
  -x509 \
  -new \
  -nodes \
  -key certs/ca/my-root-ca.key.pem \
  -days 1024 \
  -out certs/ca/my-root-ca.crt.pem \
  -subj "/C=US/ST=Utah/L=Provo/O=${FQDN}/CN=${FQDN}"
# Create a Device Certificate for each domain,
# such as example.com, *.example.com, awesome.example.com
# NOTE: You MUST match CN to the domain name or ip address you want to use
openssl genrsa \
  -out certs/server/privkey.pem \
  2048
# Create a request from your Device, which your Root CA will sign
openssl req -new \
  -key certs/server/privkey.pem \
  -out certs/tmp/csr.pem \
  -subj "/C=US/ST=Utah/L=Provo/O=${FQDN}/CN=${FQDN}"
# Sign the request from Device with your Root CA
# -CAserial certs/ca/my-root-ca.srl
openssl x509 \
  -req -in certs/tmp/csr.pem \
  -CA certs/ca/my-root-ca.crt.pem \
  -CAkey certs/ca/my-root-ca.key.pem \
  -CAcreateserial \
  -out certs/server/cert.pem \
  -days 500
# Create a public key, for funzies
# see https://gist.github.com/coolaj86/f6f36efce2821dfb046d
openssl rsa \
  -in certs/server/privkey.pem \
  -pubout -out certs/client/pubkey.pem
# Put things in their proper place
rsync -a certs/ca/my-root-ca.crt.pem certs/server/chain.pem
rsync -a certs/ca/my-root-ca.crt.pem certs/client/chain.pem
cat certs/server/cert.pem certs/server/chain.pem > certs/server/fullchain.pem
I then setup my package.json with the following:
{
  "name": "api-server",
  "version": "1.0.0",
  "description": "API Server",
  "main": "api-server.js",
  "dependencies": {
    "body-parser": "^1.15.2",
    "express": "^4.14.0"
  }
}
Ran the npm install and then created my api-server.js like this:
// Load libraries
var https      = require('https'),
    fs         = require('fs'), 
    express    = require('express'), 
    app        = express(),
    bodyParser = require('body-parser');
// Server setting
var port = process.env.PORT || 8080;
// Register body-parser
app.use(bodyParser.json());
app.use(bodyParser.urlencoded({ extended: true }));
// Configure router
var router = express.Router();
app.use('/api/v1', router);
// Register routes
router.get('/', function(req, res) {
    res.json({ success: true });
});
// Create & run https api server
var secureServer = https.createServer({
    key: fs.readFileSync('./certs/server/privkey.pem'),
    cert: fs.readFileSync('./certs/server/fullchain.pem'),
    requestCert: true,
    rejectUnauthorized: false
}, app).listen(port, function() {
    console.log('API Server Started On Port %d', port);
});
Finally, I started the app using node api-server.js and visited https://<my-ip>:8080/ in chrome.
I got the following error:
This site can’t be reached
192.168.0.21 refused to connect.
Looking on the console log of server, I saw the following:
Any ideas what I might be doing wrong here?

                        
I have found a way to solve/simply this.
make-certs.sh
api-server.js
This now works.