Thankfully, many custom ROMs for the Nexus 4, including CyanogenMod and AOKP, still include the AOSP browser as an alternative to Chrome. However, if you are running the stock Android 4.2.1 build on your Nexus 4 with root access, you can easily install the stock AOSP browser in just a few steps.
Step 1: Download the browser.apk file from here, and transfer it to your phone.
Step 2: Copy the browser.apk file to the /system/app folder. Then, long press the apk and select the option to change its ‘Permissions’. Change the permission of the APK to -rw-rw-r.
Step 3: Rename ‘BrowserProviderProxy.apk’ and ‘BrowserProviderProxy.odex’ to something. Otherwise, the AOSP browser will just force close.
Also, keep in mind that once you change the name of these files, you will not receive any OTA updates in the future. You will need to rename these files back to their original name, if you want to receive the OTA update.