Q: wifi issue with external monitor

The cause is most likely radio-frequency interference. If your router is using the 2.4 GHz radio band, switch it to the 5 GHz band, or vice versa.

From: Wi-Fi and Bluetooth: Potential sources of wireless interference - Apple Support

"Certain external monitors and LCD displays: Certain displays may emit harmonic interference, especially in the 2.4GHz band between channels 11 and 14. This interference may be at its worst if you have a portable computer with the lid closed and an external monitor connected to it. Try changing your access point to use 5 Ghz or a lower 2.4 GHz channel."