Turkish President Recep Tayyip Erdogan delivers a speech at the opening of the parliament in Ankara, Turkey, on October 1, 2018.
Ties between the two North Atlantic Treaty Organisation countries soured after Berlin criticized Ankara's crackdown on opponents following a failed 2016 coup, which saw tens of thousands arrested.
Interpretations of the controversial visit varied wildly in Turkey and Germany, with Mr Erdogan boasting it was a hugely successful but the conservative German press complaining the red carpet treatment brought nothing but hassle and expense.
"We are in favor of a four-way meeting with the presidents of Turkey, Russia and France and myself because the situation (in Syria) is still fragile", Merkel told reporters after talks with Erdogan.
"The two parties are willing to move forward to get out of the impasse", said Ilke Yoygur, an analyst at the Elcano Royal Institute in Madrid.
Relations were further strained after he was put under house arrest and Washington's decision to impose sanctions on two Turkish ministers, and double steel and aluminium tariffs in August, sparking a dramatic fall in the value of the Turkish lira.
Erdogan said Turkey does not have any claim on the rights or interests of any other country.
Have a cellphone? Expect a "presidential alert" this afternoon
The message will be broadcast by cell towers for 30 minutes, so it's possible some people may get it later than 2:18 p.m. Last week, three people filed suit in NY to block the testing of the system, known as Wireless Emergency Alerts or WEA .
"The visit's first achievement is that it took place".
"The main success of this visit is the fact that it took place", said Marc Pierini, an analyst at Carnegie Europe and former European Union ambassador to Ankara.
The snubs echoed the lukewarm welcome the Turkish leader received at a state dinner on Friday evening hosted by German President Frank-Walter Steinmeier, which several opposition politicians boycotted.
Turkey's exports also rose sharply in September, the trade ministry said, but Turkish manufacturing activity slid to its lowest level in nine years, a business survey showed.
Turkey hit back with duties on U.S. cars, alcohol and tobacco.
For Mr. Pierini, this episode "reflects the important differences between the two parties about the rule of law, in particular as regards free expression".
"It will be a long and arduous road toward normalisation", he said.