What was Colonel Douglas Mortimer's relation to the woman shown in the photograph in his watch?

Mortimer is her brother. As stated in the plot summary on Wikipedia:

While murdering the family of the man who captured him, Indio carries a musical pocketwatch that was a memento of his youth, using it to time the duel. Flashbacks reveal that he had taken the watch from a young woman (Rosemary Dexter), who had shot herself as he was raping her after having murdered her husband. The incident has haunted Indio, and he smokes an addictive drug to cloud his memory.

[...]

Mortimer takes Indio's watch and Manco remarks on Mortimer's resemblance to the woman in the photographs. Mortimer reveals himself as her brother

According to IMDb, this is the dialogue:

[Mortimer has just recovered the watch from Indio, which contains a picture of the woman that Indio raped]

Monco: [peers at the picture] There seems to be a family resemblance.

[He hands Mortimer the similar watch with the same picture he had taken earlier]

Monco: Here.

Col. Douglas Mortimer: [pause, then] Naturally, between brother and sister.

According to Howard Hughes's Once Upon a Time in the Italian West: The Filmgoers’ Guide to Spaghetti Westerns:

After Indio’s death, the dialogue between Mortimer and Manco explains the revenge motive, reinforced by Morricone’s subtle underscoring. Manco looks at the picture in the watch and notes a family resemblance. ‘Naturally,’ answers Mortimer, ‘Between brother and sister.’ Manco says that they have become rich, but for Mortimer revenge is enough. Mortimer listens to the chimes, while Morricone’s ‘Goodbye Colonel’ accompanies the scene, again incorporating the carillon, but now with lush strings and chorus. His sister avenged, Mortimer is content and rides into the sunset.

I always thought it was Indio's sister and Mortimer was the young man Indio shot. He survived and looked for revenge. She kills herself as she could not live with being raped by her brother.

It is never made clear, but the way Mortimer touches the picture of the woman in the watch at the end shows, to me, that they were not brother and sister, but lovers. You never see what image the smaller female Mortimer has in his watch, and the young man who is shot could easily be a young Mortimer.

Another perspective: Indio was the young woman's pathologically possessive and dangerously jealous brother.

In the flashback, he had shot a young Douglas Mortimer who was the lover or husband of Indio's sister. Mortimer miraculously survived Indio's attempted murder. In the disturbing rape scene, Indio's innocent sister and moral opposite could not bring herself to shoot her abusing brother. So, rather than having to live through the trauma of losing her lover, being dehumanised by an incestuous rape thus scarred for life, she shot herself.

When the man with no name asked Colonel Mortimer if he were ever young, he cryptically spoke of something that had happened which made life very precious to him. That was falling in love with Indio's sister. After surviving Indio's bullets and finding out what had happened, he was obviously on his trail but was thwarted by Indio ending up in prison. When Indio had escaped from prison and 'no name' & Colonel Mortimer saw the 'wanted' poster, the former was looking at the $10 000 reward money while the latter was alerted that Indio was at large thus reviving the trail.

In the memorable closing scenes, Indio was shocked to hear Douglas Mortimer revealing who he was and surviving the bullets he had riddled him with in that disturbing flashback. In the powerful scene when Colonel Mortimer shot Indio in the duel, he symbolically reclaimed the watch he had gifted the girl from Indio's corpse.

It's definitely not Mortimer's son who Indio shoots with the girl. The picture of the girl in the watch wouldn't mean anything to him but when he sees it and touches the picture it is with great affection. This also to me shows by the way he touches it that it's not his sister and was in fact some one he was in love with. The girl was Indio's sister. The clues are throughout the film especially when the 2 bounty Hunters meet and talk to each and Mortimer explains how life became precious and most importantly how he was young once. The look on Indio's face when Mortimer tells him his name is also a give away. The look of surprise and confusion tells you Indio knows who he is but thought he was dead.

I disagree. The young man killed by Indio was Colonel Mortimer's son. At the end of the movie he said "my boy" to Manco played by actor Clint Eastwood. The girl was Indio's sister and she killed herself for fear of becoming pregnant from an incestuous relationship.