How can I ask for compensation for Lufthansa delay in refunding me?

Did you contact Lufthansa customer service in writing (via their website contact form or email) to ask why your refund is delayed? Did you book your ticket directly with Lufthansa on lufthansa.com or did you purchase through a third-party travel agent/website? If you did not book directly with Lufthansa, they are not responsible for your refund and you should contact your travel agent instead.

Meaningful Answer:

Unless you have a higher status (SEN or HON Circle) with the airline, the only meaningful answer given your little input is that you have to fill out your request online through this web form: https://www.lufthansa.com/de/en/feedback or https://www.lufthansa.com/us/en/cancellations-and-refunds.solo_continue.

You'll get a service ticket id, and with this id, you can keep calling Lufthansa customer service in your country (although I would always call Lufthansa in Germany).

To help you on your quest for refund on this platform, you need to give us more answers:

• What was your booking reference, and what was your e-ticket number (more important, to make sure you are even eligible for refund by Lufthansa)?
• Can you post the EMD of your purchase, a listing of all the fees and the exact fare class (not Eco or Business or First, but the actual letter denoting the fare class: like V or C or F) you purchased?
• When did you buy the ticket (relevant regarding Covid-related refunding policies) and in which currency did you pay?
• Which kind of credit card did you use?
• When you received your e-ticket confirmation, what did it state regarding fare conditions and cancellation?
• How did you cancel your flight?

This information would allow me to tell you if you're actually barking up the wrong tree. Working in international legal affairs myself, I can tell you whether you might have a case.

Regarding your questions:

• Yes, you can sue Lufthansa (for what exactly?), but you won't get very far, and it's likely a very costly and lengthy process with you wasting time and money. Especially since you cancelled the flight. To start this process, you need to get yourself an international lawyer, preferably with practical airline refund claim experience. Due to this platform's policy, I can't directly recommend you one.
• Yes, you can ask for compensation (what kind of compensation and for what?) through the customer feedback form mentioned above. Allow me to tell you that you won't get any, since there is no legal base for compensation.

What might be of interest to you:

• There is the European Commission which provides a platform for Online Dispute Resolution (ODR): https://ec.europa.eu/consumers/odr/main/index.cfm?event=main.home2.show&lng=EN. Unless you're familiar with EU processes yourself, I don't recommend going that route.
• When it comes to Lufthansa and arbitration in disputes, your best option is the German SÖP ("Schlichtungsstelle für den öffentlichen Personennahverkehr"): https://soep-online.de/en/. They will ask you the same questions I have.

On a sidenote:

Since you cancelled your flight, you have no rights as defined in Regulation (EC) No 261/2004. You will also have no case with the NEB (National Enforcement Bodies) https://transport.ec.europa.eu/transport-themes/passenger-rights/national-enforcement-bodies-neb_en of the European Union.

If you currently go onto the Lufthansa FAQ site regarding cancellations, you get the following answer to the question of: "What is the status of my refund?"

I hope this helps.